ITP001 Axioms: ITP135^7.ax
%------------------------------------------------------------------------------
% File : ITP135^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : complex.ax [Gau19]
% : HL4135^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 311 ( 173 unt; 61 typ; 0 def)
% Number of atoms : 399 ( 329 equ; 48 cnn)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 2793 ( 48 ~; 9 |; 20 &;2636 @)
% ( 41 <=>; 39 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg;2636 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 84 ( 84 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 58 usr; 5 con; 0-4 aty)
% Number of variables : 460 ( 0 ^ 449 !; 1 ?; 460 :)
% ( 10 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Erealax_2Ereal,type,
tyop_2Erealax_2Ereal: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ereal_2E_2F,type,
c_2Ereal_2E_2F: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Ecomplex_2EIM,type,
c_2Ecomplex_2EIM: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ecomplex_2ERE,type,
c_2Ecomplex_2ERE: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ereal_2Eabs,type,
c_2Ereal_2Eabs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Eacs,type,
c_2Etransc_2Eacs: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ecomplex_2Earg,type,
c_2Ecomplex_2Earg: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Easn,type,
c_2Etransc_2Easn: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ecomplex_2Ecomplex__add,type,
c_2Ecomplex_2Ecomplex__add: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__div,type,
c_2Ecomplex_2Ecomplex__div: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__exp,type,
c_2Ecomplex_2Ecomplex__exp: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__inv,type,
c_2Ecomplex_2Ecomplex__inv: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__mul,type,
c_2Ecomplex_2Ecomplex__mul: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__neg,type,
c_2Ecomplex_2Ecomplex__neg: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__of__num,type,
c_2Ecomplex_2Ecomplex__of__num: tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__of__real,type,
c_2Ecomplex_2Ecomplex__of__real: tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__pow,type,
c_2Ecomplex_2Ecomplex__pow: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Enum_2Enum > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__scalar__lmul,type,
c_2Ecomplex_2Ecomplex__scalar__lmul: tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__scalar__rmul,type,
c_2Ecomplex_2Ecomplex__scalar__rmul: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Ecomplex__sub,type,
c_2Ecomplex_2Ecomplex__sub: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Ecomplex_2Econj,type,
c_2Ecomplex_2Econj: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) ).
thf(c_2Etransc_2Ecos,type,
c_2Etransc_2Ecos: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Eexp,type,
c_2Etransc_2Eexp: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ecomplex_2Ei,type,
c_2Ecomplex_2Ei: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Einv,type,
c_2Erealax_2Einv: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ecomplex_2Emodu,type,
c_2Ecomplex_2Emodu: ( tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal ) > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Epi,type,
c_2Etransc_2Epi: tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Epow,type,
c_2Ereal_2Epow: tyop_2Erealax_2Ereal > tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__add,type,
c_2Erealax_2Ereal__add: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__lt,type,
c_2Erealax_2Ereal__lt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Ereal_2Ereal__lte,type,
c_2Ereal_2Ereal__lte: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > $o ).
thf(c_2Erealax_2Ereal__mul,type,
c_2Erealax_2Ereal__mul: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Erealax_2Ereal__neg,type,
c_2Erealax_2Ereal__neg: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__of__num,type,
c_2Ereal_2Ereal__of__num: tyop_2Enum_2Enum > tyop_2Erealax_2Ereal ).
thf(c_2Ereal_2Ereal__sub,type,
c_2Ereal_2Ereal__sub: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Esin,type,
c_2Etransc_2Esin: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Etransc_2Esqrt,type,
c_2Etransc_2Esqrt: tyop_2Erealax_2Ereal > tyop_2Erealax_2Ereal ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ecomplex_2ERE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2ERE @ V0z )
= ( c_2Epair_2EFST @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0z ) ) ).
thf(thm_2Ecomplex_2EIM,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2EIM @ V0z )
= ( c_2Epair_2ESND @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0z ) ) ).
thf(thm_2Ecomplex_2Ecomplex__of__real,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__of__real @ V0x )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ V0x @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__of__num,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__of__num @ V0n )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2Ereal__of__num @ V0n ) ) ) ).
thf(thm_2Ecomplex_2Ei,axiom,
( c_2Ecomplex_2Ei
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__add,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__add @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2ERE @ V1w ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2EIM @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__neg,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ V0z )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__neg @ ( c_2Ecomplex_2ERE @ V0z ) ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ecomplex_2EIM @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__mul,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2ERE @ V1w ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2EIM @ V1w ) ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2EIM @ V1w ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2ERE @ V1w ) ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__inv,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__inv @ V0z )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Erealax_2Ereal__neg @ ( c_2Ecomplex_2EIM @ V0z ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__sub,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__add @ V0z @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__div,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__inv @ V1w ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__scalar__lmul,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ V0k @ ( c_2Ecomplex_2ERE @ V1z ) ) @ ( c_2Erealax_2Ereal__mul @ V0k @ ( c_2Ecomplex_2EIM @ V1z ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__scalar__rmul,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1k: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V0z @ V1k )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2ERE @ V0z ) @ V1k ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2EIM @ V0z ) @ V1k ) ) ) ).
thf(thm_2Ecomplex_2Econj,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ V0z )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Erealax_2Ereal__neg @ ( c_2Ecomplex_2EIM @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2Emodu,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ V0z )
= ( c_2Etransc_2Esqrt @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2Earg,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Earg @ V0z )
= ( c_2Ebool_2ECOND @ tyop_2Erealax_2Ereal @ ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ecomplex_2EIM @ V0z ) ) @ ( c_2Etransc_2Eacs @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__neg @ ( c_2Etransc_2Eacs @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ) @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ c_2Etransc_2Epi ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__pow__def,axiom,
( ! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__pow @ V0z @ c_2Enum_2E0 )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ V1z @ ( c_2Enum_2ESUC @ V2n ) )
= ( c_2Ecomplex_2Ecomplex__mul @ V1z @ ( c_2Ecomplex_2Ecomplex__pow @ V1z @ V2n ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__exp,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__exp @ V0z )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Etransc_2Eexp @ ( c_2Ecomplex_2ERE @ V0z ) ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Ecomplex_2EIM @ V0z ) ) @ ( c_2Etransc_2Esin @ ( c_2Ecomplex_2EIM @ V0z ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__LEMMA1,axiom,
! [V0a: tyop_2Erealax_2Ereal,V1b: tyop_2Erealax_2Ereal,V2c: tyop_2Erealax_2Ereal,V3d: tyop_2Erealax_2Ereal] :
( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ereal_2Ereal__sub @ ( c_2Erealax_2Ereal__mul @ V0a @ V2c ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V3d ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Erealax_2Ereal__add @ ( c_2Erealax_2Ereal__mul @ V0a @ V3d ) @ ( c_2Erealax_2Ereal__mul @ V1b @ V2c ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ V0a @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V1b @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ V2c @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V3d @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__LEMMA2,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ V0x ) @ ( c_2Etransc_2Esqrt @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ V0x @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ V1y @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2EIM @ V0z ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__RE__IM__EQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( V0z = V1w )
<=> ( ( ( c_2Ecomplex_2ERE @ V0z )
= ( c_2Ecomplex_2ERE @ V1w ) )
& ( ( c_2Ecomplex_2EIM @ V0z )
= ( c_2Ecomplex_2EIM @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2ERE__COMPLEX__OF__REAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2ERE @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) )
= V0x ) ).
thf(thm_2Ecomplex_2EIM__COMPLEX__OF__REAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2EIM @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__0,axiom,
( ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__1,axiom,
( ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__10,axiom,
( (~)
@ ( ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__0__THM,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__COMM,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__add @ V1w @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__ASSOC,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V0z @ ( c_2Ecomplex_2Ecomplex__add @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) @ V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__RID,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V0z @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__LID,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) @ V0z )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__RINV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V0z @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__LINV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__COMM,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__ASSOC,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) @ V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__RID,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__LID,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0z )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__RINV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__LINV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__LDISTRIB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__add @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__RDISTRIB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) @ V2v )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__LADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__add @ V0z @ V2v ) )
<=> ( V1w = V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__RADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V2v )
= ( c_2Ecomplex_2Ecomplex__add @ V1w @ V2v ) )
<=> ( V0z = V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__RID__UNIQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= V0z )
<=> ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__LID__UNIQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= V1w )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEGNEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__EQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__neg @ V0z )
= V1w )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__NEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__neg @ V0z )
= ( c_2Ecomplex_2Ecomplex__neg @ V1w ) )
<=> ( V0z = V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__RNEG__UNIQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( V1w
= ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__LNEG__UNIQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__ADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__RZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__LZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__LMUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__RMUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__MUL2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ENTIRE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
| ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__0,axiom,
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__EQ0,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__neg @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__REFL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__RZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__LZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__LNEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ V1w )
= ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__NEG2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__RNEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__ADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) @ V1w )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__ADD2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V1w @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) @ V0z )
= V1w ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__SUB2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= V1w ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__SUB2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD2__SUB2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2u: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V3v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__add @ V2u @ V3v ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V2u ) @ ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V3v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__TRIANGLE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__0,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( V0z = V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__SUB__LADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( V0z
= ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V2v ) )
<=> ( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V2v )
= V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__SUB__RADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w )
= V2v )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__add @ V2v @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__RNEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__LNEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ V1w )
= ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__LDISTRIB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__RDISTRIB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) @ V2v )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIFFSQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V0z ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__LMUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) )
<=> ( ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
| ( V1w = V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__RMUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v )
= ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2v ) )
<=> ( ( V2v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
| ( V0z = V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__LMUL2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( V1w = V2v )
<=> ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__RMUL__IMP,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V0z )
= ( c_2Ecomplex_2Ecomplex__mul @ V2v @ V0z ) ) )
=> ( V1w = V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__LMUL__IMP,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) ) )
=> ( V1w = V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__INV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__MUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) @ ( c_2Ecomplex_2Ecomplex__inv @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INVINV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) )
= V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__LINV__UNIQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( V0z
= ( c_2Ecomplex_2Ecomplex__inv @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__RINV__UNIQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
=> ( V1w
= ( c_2Ecomplex_2Ecomplex__inv @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__0,axiom,
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV1,axiom,
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__INV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__NEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__EQ__0,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__inv @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__NZ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( (~)
@ ( ( c_2Ecomplex_2Ecomplex__inv @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__INJ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__inv @ V0z )
= ( c_2Ecomplex_2Ecomplex__inv @ V1w ) )
<=> ( V0z = V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__LDIV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__RDIV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__div @ V0z @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__DIV2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) @ ( c_2Ecomplex_2Ecomplex__neg @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__1OVER,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__inv @ V0z )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV1,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ V0z @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__ADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V2v ) @ ( c_2Ecomplex_2Ecomplex__div @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) @ V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V2v ) @ ( c_2Ecomplex_2Ecomplex__div @ V1w @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) @ V2v ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__RMUL__CANCEL,axiom,
! [V0v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__mul @ V1z @ V0v ) @ ( c_2Ecomplex_2Ecomplex__mul @ V2w @ V0v ) )
= ( c_2Ecomplex_2Ecomplex__div @ V1z @ V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__LMUL__CANCEL,axiom,
! [V0v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__mul @ V0v @ V1z ) @ ( c_2Ecomplex_2Ecomplex__mul @ V0v @ V2w ) )
= ( c_2Ecomplex_2Ecomplex__div @ V1z @ V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__DENOM__CANCEL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__div @ V1w @ V0z ) @ ( c_2Ecomplex_2Ecomplex__div @ V2v @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__div @ V1w @ V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__INNER__CANCEL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__div @ V1w @ V0z ) @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V2v ) )
= ( c_2Ecomplex_2Ecomplex__div @ V1w @ V2v ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__OUTER__CANCEL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__div @ V2v @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__div @ V2v @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__MUL2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ! [V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ V2v @ V1w )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v ) @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__RAT,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2u: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V3v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V3v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__div @ V2u @ V3v ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V3v ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2u ) ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V3v ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__RAT,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2u: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V3v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V3v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) @ ( c_2Ecomplex_2Ecomplex__div @ V2u @ V3v ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V3v ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2u ) ) @ ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V3v ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__LZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__REFL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__div @ V0z @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__INV2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) @ ( c_2Ecomplex_2Ecomplex__inv @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V0z ) @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__RDIV__EQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V2v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( V0z
= ( c_2Ecomplex_2Ecomplex__div @ V1w @ V2v ) )
<=> ( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V2v )
= V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__LDIV__EQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2v: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V2v
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( ( c_2Ecomplex_2Ecomplex__div @ V0z @ V2v )
= V1w )
<=> ( V0z
= ( c_2Ecomplex_2Ecomplex__mul @ V1w @ V2v ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__EQ,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__of__real @ V0x )
= ( c_2Ecomplex_2Ecomplex__of__real @ V1y ) )
<=> ( V0x = V1y ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__ADD,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) @ ( c_2Ecomplex_2Ecomplex__of__real @ V1y ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__NEG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__MUL,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) @ ( c_2Ecomplex_2Ecomplex__of__real @ V1y ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Erealax_2Ereal__mul @ V0x @ V1y ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__INV,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Erealax_2Einv @ V0x ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__SUB,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) @ ( c_2Ecomplex_2Ecomplex__of__real @ V1y ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__REAL__DIV,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) @ ( c_2Ecomplex_2Ecomplex__of__real @ V1y ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2E_2F @ V0x @ V1y ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__NUM__EQ,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( ( c_2Ecomplex_2Ecomplex__of__num @ V0m )
= ( c_2Ecomplex_2Ecomplex__of__num @ V1n ) )
<=> ( V0m = V1n ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__NUM__ADD,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__of__num @ V0m ) @ ( c_2Ecomplex_2Ecomplex__of__num @ V1n ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2E_2B @ V0m @ V1n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__OF__NUM__MUL,axiom,
! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__of__num @ V0m ) @ ( c_2Ecomplex_2Ecomplex__of__num @ V1n ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2E_2A @ V0m @ V1n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1l @ V2z ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Ereal__mul @ V0k @ V1l ) @ V2z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Ereal__neg @ V0k ) @ V1z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__neg @ V1z ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Ereal__neg @ V0k ) @ V1z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ADD,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Ereal__add @ V0k @ V1l ) @ V2z )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2z ) @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1l @ V2z ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__SUB,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Ereal__sub @ V0k @ V1l ) @ V2z )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2z ) @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1l @ V2z ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__add @ V1z @ V2w ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__sub @ V1z @ V2w ) )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__SCALAR__LMUL2,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V3w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2z ) @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1l @ V3w ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Ereal__mul @ V0k @ V1l ) @ ( c_2Ecomplex_2Ecomplex__mul @ V2z @ V3w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__LMUL__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) @ V2w )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__mul @ V1z @ V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__RMUL__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V1z @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2w ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__mul @ V1z @ V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ONE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0z )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__NEG1,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DOUBLE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__add @ V0z @ V0z )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__ENTIRE,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( ( V0k
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V1z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EQ__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2w ) )
<=> ( ( V0k
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
| ( V1z = V2w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2z )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1l @ V2z ) )
<=> ( ( V0k = V1l )
| ( V2z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__EQ1,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z )
= V1z )
<=> ( ( V0k
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
| ( V1z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V0k
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V1z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Erealax_2Einv @ V0k ) @ ( c_2Ecomplex_2Ecomplex__inv @ V1z ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__LMUL__DIV2,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V3w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( (~)
@ ( V1l
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) )
& ( (~)
@ ( V3w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) )
=> ( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V2z ) @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1l @ V3w ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2E_2F @ V0k @ V1l ) @ ( c_2Ecomplex_2Ecomplex__div @ V2z @ V3w ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__MUL__COMM,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z )
= ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V1z @ V0k ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ V0k ) @ V1l )
= ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ ( c_2Erealax_2Ereal__mul @ V0k @ V1l ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V1z @ V0k ) )
= ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V1z @ ( c_2Erealax_2Ereal__neg @ V0k ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__NEG__SCALAR__RMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ ( c_2Ecomplex_2Ecomplex__neg @ V1z ) @ V0k )
= ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V1z @ ( c_2Erealax_2Ereal__neg @ V0k ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ADD,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ ( c_2Erealax_2Ereal__add @ V0k @ V1l ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ V0k ) @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ V1l ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__RSCALAR__RMUL__SUB,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1l: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ ( c_2Ereal_2Ereal__sub @ V0k @ V1l ) )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ V0k ) @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2z @ V1l ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ADD__RSCALAR__RMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ ( c_2Ecomplex_2Ecomplex__add @ V1z @ V2w ) @ V0k )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V1z @ V0k ) @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2w @ V0k ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SUB__SCALAR__RMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ ( c_2Ecomplex_2Ecomplex__sub @ V1z @ V2w ) @ V0k )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V1z @ V0k ) @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V2w @ V0k ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V0z @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__ZERO__SCALAR__RMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) @ V0k )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__ONE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V0z @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__SCALAR__RMUL__NEG1,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__rmul @ V0z @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
= ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) ).
thf(thm_2Ecomplex_2ECONJ__REAL__REFL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) ) ).
thf(thm_2Ecomplex_2ECONJ__NUM__REFL,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__of__num @ V0n ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ V0n ) ) ).
thf(thm_2Ecomplex_2ECONJ__ADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__add @ ( c_2Ecomplex_2Econj @ V0z ) @ ( c_2Ecomplex_2Econj @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__NEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__neg @ ( c_2Ecomplex_2Econj @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__sub @ ( c_2Ecomplex_2Econj @ V0z ) @ ( c_2Ecomplex_2Econj @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__MUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Econj @ V0z ) @ ( c_2Ecomplex_2Econj @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__INV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Econj @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__DIV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Econj @ V0z ) @ ( c_2Ecomplex_2Econj @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ ( c_2Ecomplex_2Econj @ V1z ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__CONJ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Econj @ V0z ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECONJ__EQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Econj @ V0z )
= V1w )
<=> ( V0z
= ( c_2Ecomplex_2Econj @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__EQ2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Econj @ V0z )
= ( c_2Ecomplex_2Econj @ V1w ) )
<=> ( V0z = V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Econj @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Econj @ V0z ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECONJ__ZERO,axiom,
( ( c_2Ecomplex_2Econj @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2EMODU__POW2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Epow @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Erealax_2Ereal__add @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2ERE__IM__LE__MODU,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ecomplex_2ERE @ V0z ) ) @ ( c_2Ecomplex_2Emodu @ V0z ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Eabs @ ( c_2Ecomplex_2EIM @ V0z ) ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ).
thf(thm_2Ecomplex_2EMODU__POS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] : ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__RCONJ1,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ V0z @ ( c_2Ecomplex_2Econj @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__LCONJ1,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Econj @ V0z ) @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__real @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(thm_2Ecomplex_2EMODU__NEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= ( c_2Ecomplex_2Emodu @ V0z ) ) ).
thf(thm_2Ecomplex_2EMODU__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__sub @ V1w @ V0z ) ) ) ).
thf(thm_2Ecomplex_2EMODU__CONJ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Econj @ V0z ) )
= ( c_2Ecomplex_2Emodu @ V0z ) ) ).
thf(thm_2Ecomplex_2EMODU__MUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V1w ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Ecomplex_2Emodu @ V1w ) ) ) ).
thf(thm_2Ecomplex_2EMODU__0,axiom,
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2EMODU__1,axiom,
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecomplex_2EMODU__COMPLEX__INV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) )
= ( c_2Erealax_2Einv @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2EMODU__DIV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V1w
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) )
= ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Ecomplex_2Emodu @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2EMODU__SCALAR__LMUL,axiom,
! [V0k: tyop_2Erealax_2Ereal,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V0k @ V1z ) )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Eabs @ V0k ) @ ( c_2Ecomplex_2Emodu @ V1z ) ) ) ).
thf(thm_2Ecomplex_2EMODU__REAL,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__of__real @ V0x ) )
= ( c_2Ereal_2Eabs @ V0x ) ) ).
thf(thm_2Ecomplex_2EMODU__NUM,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__of__num @ V0n ) )
= ( c_2Ereal_2Ereal__of__num @ V0n ) ) ).
thf(thm_2Ecomplex_2EMODU__ZERO,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( ( c_2Ecomplex_2Emodu @ V0z )
= ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2EMODU__NZ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
<=> ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ).
thf(thm_2Ecomplex_2EMODU__CASES,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
| ( c_2Erealax_2Ereal__lt @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ERE__DIV__MODU__BOUNDS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2EIM__DIV__MODU__BOUNDS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ERE__DIV__MODU__ACS__BOUNDS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Ereal_2Ereal__of__num @ c_2Enum_2E0 ) @ ( c_2Etransc_2Eacs @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Eacs @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) @ c_2Etransc_2Epi ) ) ) ).
thf(thm_2Ecomplex_2EIM__DIV__MODU__ASN__BOUNDS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Ereal_2Ereal__lte @ ( c_2Erealax_2Ereal__neg @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) @ ( c_2Etransc_2Easn @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) )
& ( c_2Ereal_2Ereal__lte @ ( c_2Etransc_2Easn @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) @ ( c_2Ereal_2E_2F @ c_2Etransc_2Epi @ ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ERE__DIV__MODU__ACS__COS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Etransc_2Ecos @ ( c_2Etransc_2Eacs @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2EIM__DIV__MODU__ASN__SIN,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Etransc_2Esin @ ( c_2Etransc_2Easn @ ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) )
= ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2EARG__COS,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Etransc_2Ecos @ ( c_2Ecomplex_2Earg @ V0z ) )
= ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2ERE @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2EARG__SIN,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( ( c_2Etransc_2Esin @ ( c_2Ecomplex_2Earg @ V0z ) )
= ( c_2Ereal_2E_2F @ ( c_2Ecomplex_2EIM @ V0z ) @ ( c_2Ecomplex_2Emodu @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2ERE__MODU__ARG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2ERE @ V0z )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Etransc_2Ecos @ ( c_2Ecomplex_2Earg @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2EIM__MODU__ARG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2EIM @ V0z )
= ( c_2Erealax_2Ereal__mul @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Etransc_2Esin @ ( c_2Ecomplex_2Earg @ V0z ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__TRIANGLE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Ecomplex_2Earg @ V0z ) ) @ ( c_2Etransc_2Esin @ ( c_2Ecomplex_2Earg @ V0z ) ) ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MODU__ARG__EQ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( V0z = V1w )
<=> ( ( ( c_2Ecomplex_2Emodu @ V0z )
= ( c_2Ecomplex_2Emodu @ V1w ) )
& ( ( c_2Ecomplex_2Earg @ V0z )
= ( c_2Ecomplex_2Earg @ V1w ) ) ) ) ).
thf(thm_2Ecomplex_2EMODU__UNIT,axiom,
! [A_27a: $tType,V0x: tyop_2Erealax_2Ereal,V1y: A_27a] :
( ( c_2Ecomplex_2Emodu @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V0x ) ) )
= ( c_2Ereal_2Ereal__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__MUL__ARG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V0x ) ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V1y ) @ ( c_2Etransc_2Esin @ V1y ) ) )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) @ ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__add @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__INV__ARG,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V0x ) ) )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__neg @ V0x ) ) @ ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__neg @ V0x ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__DIV__ARG,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1y: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__div @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V0x ) ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V1y ) @ ( c_2Etransc_2Esin @ V1y ) ) )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) @ ( c_2Etransc_2Esin @ ( c_2Ereal_2Ereal__sub @ V0x @ V1y ) ) ) ) ).
thf(thm_2Ecomplex_2Ecomplex__pow__def__compute,axiom,
( ! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__pow @ V0z @ c_2Enum_2E0 )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) )
& ! [V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ V1z @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) )
= ( c_2Ecomplex_2Ecomplex__mul @ V1z @ ( c_2Ecomplex_2Ecomplex__pow @ V1z @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V2n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ) )
& ! [V3z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V4n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ V3z @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V4n ) ) )
= ( c_2Ecomplex_2Ecomplex__mul @ V3z @ ( c_2Ecomplex_2Ecomplex__pow @ V3z @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V4n ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__0,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__NZ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ( (~)
@ ( ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V1n )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POWINV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( (~)
@ ( V0z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) )
=> ! [V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V1n ) )
= ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) @ V1n ) ) ) ).
thf(thm_2Ecomplex_2EMODU__COMPLEX__POW,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Emodu @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V1n ) )
= ( c_2Ereal_2Epow @ ( c_2Ecomplex_2Emodu @ V0z ) @ V1n ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__ADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ V0z @ ( c_2Earithmetic_2E_2B @ V1m @ V2n ) )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V1m ) @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V2n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__1,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__pow @ V0z @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__pow @ V0z @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) )
= ( c_2Ecomplex_2Ecomplex__mul @ V0z @ V0z ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__ONE,axiom,
! [V0n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0n )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__MUL,axiom,
! [V0n: tyop_2Enum_2Enum,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__mul @ V1z @ V2w ) @ V0n )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__pow @ V1z @ V0n ) @ ( c_2Ecomplex_2Ecomplex__pow @ V2w @ V0n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__INV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__inv @ V0z ) @ V1n )
= ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V1n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__DIV,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V2n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__div @ V0z @ V1w ) @ V2n )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V2n ) @ ( c_2Ecomplex_2Ecomplex__pow @ V1w @ V2n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__L,axiom,
! [V0n: tyop_2Enum_2Enum,V1k: tyop_2Erealax_2Ereal,V2z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ V1k @ V2z ) @ V0n )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Epow @ V1k @ V0n ) @ ( c_2Ecomplex_2Ecomplex__pow @ V2z @ V0n ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__ZERO,axiom,
! [V0n: tyop_2Enum_2Enum,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__pow @ V1z @ V0n )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
=> ( V1z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__ZERO__EQ,axiom,
! [V0n: tyop_2Enum_2Enum,V1z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( ( c_2Ecomplex_2Ecomplex__pow @ V1z @ ( c_2Enum_2ESUC @ V0n ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
<=> ( V1z
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__POW__POW,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1m: tyop_2Enum_2Enum,V2n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__pow @ V0z @ V1m ) @ V2n )
= ( c_2Ecomplex_2Ecomplex__pow @ V0z @ ( c_2Earithmetic_2E_2A @ V1m @ V2n ) ) ) ).
thf(thm_2Ecomplex_2EDE__MOIVRE__LEMMA,axiom,
! [V0x: tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V0x ) ) @ V1n )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0x ) ) @ ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0x ) ) ) ) ).
thf(thm_2Ecomplex_2EDE__MOIVRE__THM,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Ecomplex_2Earg @ V0z ) ) @ ( c_2Etransc_2Esin @ ( c_2Ecomplex_2Earg @ V0z ) ) ) ) @ V1n )
= ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Epow @ ( c_2Ecomplex_2Emodu @ V0z ) @ V1n ) @ ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ecomplex_2Earg @ V0z ) ) ) @ ( c_2Etransc_2Esin @ ( c_2Erealax_2Ereal__mul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ ( c_2Ecomplex_2Earg @ V0z ) ) ) ) ) ) ).
thf(thm_2Ecomplex_2EEXP__IMAGINARY,axiom,
! [V0x: tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ c_2Ecomplex_2Ei @ V0x ) )
= ( c_2Epair_2E_2C @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal @ ( c_2Etransc_2Ecos @ V0x ) @ ( c_2Etransc_2Esin @ V0x ) ) ) ).
thf(thm_2Ecomplex_2EEULER__FORMULE,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ecomplex_2Emodu @ V0z ) @ ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__scalar__rmul @ c_2Ecomplex_2Ei @ ( c_2Ecomplex_2Earg @ V0z ) ) ) )
= V0z ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__ADD,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) @ ( c_2Ecomplex_2Ecomplex__exp @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__NEG,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__inv @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__SUB,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__sub @ V0z @ V1w ) )
= ( c_2Ecomplex_2Ecomplex__div @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) @ ( c_2Ecomplex_2Ecomplex__exp @ V1w ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__N,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__scalar__lmul @ ( c_2Ereal_2Ereal__of__num @ V1n ) @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) @ V1n ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__N2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1n: tyop_2Enum_2Enum] :
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__of__num @ V1n ) @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__pow @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) @ V1n ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__0,axiom,
( ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__NZ,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( (~)
@ ( ( c_2Ecomplex_2Ecomplex__exp @ V0z )
= ( c_2Ecomplex_2Ecomplex__of__num @ c_2Enum_2E0 ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__ADD__MUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal,V1w: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__add @ V0z @ V1w ) ) @ ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) )
= ( c_2Ecomplex_2Ecomplex__exp @ V1w ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) @ ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Ecomplex_2ECOMPLEX__EXP__NEG__MUL2,axiom,
! [V0z: tyop_2Epair_2Eprod @ tyop_2Erealax_2Ereal @ tyop_2Erealax_2Ereal] :
( ( c_2Ecomplex_2Ecomplex__mul @ ( c_2Ecomplex_2Ecomplex__exp @ ( c_2Ecomplex_2Ecomplex__neg @ V0z ) ) @ ( c_2Ecomplex_2Ecomplex__exp @ V0z ) )
= ( c_2Ecomplex_2Ecomplex__of__num @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
%------------------------------------------------------------------------------