SET007 Axioms: SET007+666.ax
%------------------------------------------------------------------------------
% File : SET007+666 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Trigonometric Form of Complex Numbers
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : comptrig [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 84 ( 24 unt; 0 def)
% Number of atoms : 285 ( 55 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 273 ( 72 ~; 2 |; 77 &)
% ( 7 <=>; 115 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 46 ( 46 usr; 12 con; 0-6 aty)
% Number of variables : 95 ( 93 !; 2 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(t1_comptrig,axiom,
$true ).
fof(t2_comptrig,axiom,
$true ).
fof(t3_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> r1_xreal_0(k1_real_1(k17_complex1(A)),k3_complex1(A)) ) ).
fof(t4_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> r1_xreal_0(k1_real_1(k17_complex1(A)),k4_complex1(A)) ) ).
fof(t5_comptrig,axiom,
$true ).
fof(t6_comptrig,axiom,
$true ).
fof(t7_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> k7_square_1(k17_complex1(A)) = k3_real_1(k7_square_1(k3_complex1(A)),k7_square_1(k4_complex1(A))) ) ).
fof(t8_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( k1_hahnban1(A,B) = k1_hahnban1(C,D)
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t9_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> A = k1_hahnban1(k3_complex1(A),k4_complex1(A)) ) ).
fof(t10_comptrig,axiom,
$true ).
fof(t11_comptrig,axiom,
$true ).
fof(t12_comptrig,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,np__1) = B ) ) ).
fof(t13_comptrig,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,np__2) = k1_group_1(A,B,B) ) ) ).
fof(t14_comptrig,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ) ).
fof(t15_comptrig,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),k10_group_1(A,B,C),D) = k10_group_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,D),k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,D)) ) ) ) ) ).
fof(t16_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),k1_hahnban1(A,np__0),B) = k1_hahnban1(k4_power(A,B),np__0) ) ) ) ).
fof(t17_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__0,A)
=> ( B = np__0
| k3_power(k1_power(B,A),B) = A ) ) ) ) ).
fof(t18_comptrig,axiom,
$true ).
fof(t19_comptrig,axiom,
$true ).
fof(t20_comptrig,axiom,
( k3_real_1(k32_sin_cos,k6_real_1(k32_sin_cos,np__2)) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos)
& k3_real_1(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k6_real_1(k32_sin_cos,np__2)) = k4_real_1(np__2,k32_sin_cos)
& k5_real_1(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k32_sin_cos) = k6_real_1(k32_sin_cos,np__2) ) ).
fof(t21_comptrig,axiom,
( ~ r1_xreal_0(k6_real_1(k32_sin_cos,np__2),np__0)
& ~ r1_xreal_0(k32_sin_cos,k6_real_1(k32_sin_cos,np__2))
& ~ r1_xreal_0(k32_sin_cos,np__0)
& ~ r1_xreal_0(k6_real_1(k32_sin_cos,np__2),k1_real_1(k6_real_1(k32_sin_cos,np__2)))
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),k32_sin_cos)
& ~ r1_xreal_0(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k6_real_1(k32_sin_cos,np__2))
& ~ r1_xreal_0(np__0,k1_real_1(k6_real_1(k32_sin_cos,np__2)))
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),np__0)
& ~ r1_xreal_0(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k32_sin_cos)
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos))
& ~ r1_xreal_0(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),np__0) ) ).
fof(t22_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ~ ( r2_hidden(D,k2_rcomp_1(A,C))
& ~ r2_hidden(D,k2_rcomp_1(A,B))
& D != B
& ~ r2_hidden(D,k2_rcomp_1(B,C)) ) ) ) ) ) ).
fof(t23_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(np__0,k32_sin_cos))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),np__0) ) ) ).
fof(t24_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_rcomp_1(np__0,k32_sin_cos))
=> r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)) ) ) ).
fof(t25_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(k32_sin_cos,k4_real_1(np__2,k32_sin_cos)))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A)) ) ) ).
fof(t26_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_rcomp_1(k32_sin_cos,k4_real_1(np__2,k32_sin_cos)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),np__0) ) ) ).
fof(t27_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(k1_real_1(k6_real_1(k32_sin_cos,np__2)),k6_real_1(k32_sin_cos,np__2)))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),np__0) ) ) ).
fof(t28_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_rcomp_1(k1_real_1(k6_real_1(k32_sin_cos,np__2)),k6_real_1(k32_sin_cos,np__2)))
=> r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) ) ) ).
fof(t29_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(k6_real_1(k32_sin_cos,np__2),k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos)))
& r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) ) ) ).
fof(t30_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_rcomp_1(k6_real_1(k32_sin_cos,np__2),k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos)))
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),np__0) ) ) ).
fof(t31_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r2_hidden(A,k2_rcomp_1(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k4_real_1(np__2,k32_sin_cos)))
& r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),np__0) ) ) ).
fof(t32_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(A,k1_rcomp_1(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k4_real_1(np__2,k32_sin_cos)))
=> r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A)) ) ) ).
fof(t33_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),A)
& k19_sin_cos(A) = np__0
& A != np__0
& A != k32_sin_cos ) ) ).
fof(t34_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),A)
& k22_sin_cos(A) = np__0
& A != k6_real_1(k32_sin_cos,np__2)
& A != k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) ) ) ).
fof(t35_comptrig,axiom,
r1_rfunct_2(k18_sin_cos,k2_rcomp_1(k1_real_1(k6_real_1(k32_sin_cos,np__2)),k6_real_1(k32_sin_cos,np__2))) ).
fof(t36_comptrig,axiom,
r2_rfunct_2(k18_sin_cos,k2_rcomp_1(k6_real_1(k32_sin_cos,np__2),k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos))) ).
fof(t37_comptrig,axiom,
r2_rfunct_2(k21_sin_cos,k2_rcomp_1(np__0,k32_sin_cos)) ).
fof(t38_comptrig,axiom,
r1_rfunct_2(k21_sin_cos,k2_rcomp_1(k32_sin_cos,k4_real_1(np__2,k32_sin_cos))) ).
fof(t39_comptrig,axiom,
r1_rfunct_2(k18_sin_cos,k1_rcomp_1(k1_real_1(k6_real_1(k32_sin_cos,np__2)),k6_real_1(k32_sin_cos,np__2))) ).
fof(t40_comptrig,axiom,
r2_rfunct_2(k18_sin_cos,k1_rcomp_1(k6_real_1(k32_sin_cos,np__2),k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos))) ).
fof(t41_comptrig,axiom,
r2_rfunct_2(k21_sin_cos,k1_rcomp_1(np__0,k32_sin_cos)) ).
fof(t42_comptrig,axiom,
r1_rfunct_2(k21_sin_cos,k1_rcomp_1(k32_sin_cos,k4_real_1(np__2,k32_sin_cos))) ).
fof(t43_comptrig,axiom,
( r2_fcont_1(k18_sin_cos,k1_numbers)
& ! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r2_fcont_1(k18_sin_cos,k1_rcomp_1(A,B))
& r2_fcont_1(k18_sin_cos,k2_rcomp_1(A,B)) ) ) ) ) ).
fof(t44_comptrig,axiom,
( r2_fcont_1(k21_sin_cos,k1_numbers)
& ! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r2_fcont_1(k21_sin_cos,k1_rcomp_1(A,B))
& r2_fcont_1(k21_sin_cos,k2_rcomp_1(A,B)) ) ) ) ) ).
fof(t45_comptrig,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r2_hidden(k2_seq_1(k1_numbers,k1_numbers,k18_sin_cos,A),k1_rcomp_1(k1_real_1(np__1),np__1))
& r2_hidden(k2_seq_1(k1_numbers,k1_numbers,k21_sin_cos,A),k1_rcomp_1(k1_real_1(np__1),np__1)) ) ) ).
fof(t46_comptrig,axiom,
k2_relat_1(k18_sin_cos) = k1_rcomp_1(k1_real_1(np__1),np__1) ).
fof(t47_comptrig,axiom,
k2_relat_1(k21_sin_cos) = k1_rcomp_1(k1_real_1(np__1),np__1) ).
fof(t48_comptrig,axiom,
k2_relat_1(k2_partfun1(k1_numbers,k1_numbers,k18_sin_cos,k1_rcomp_1(k1_real_1(k6_real_1(k32_sin_cos,np__2)),k6_real_1(k32_sin_cos,np__2)))) = k1_rcomp_1(k1_real_1(np__1),np__1) ).
fof(t49_comptrig,axiom,
k2_relat_1(k2_partfun1(k1_numbers,k1_numbers,k18_sin_cos,k1_rcomp_1(k6_real_1(k32_sin_cos,np__2),k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos)))) = k1_rcomp_1(k1_real_1(np__1),np__1) ).
fof(t50_comptrig,axiom,
k2_relat_1(k2_partfun1(k1_numbers,k1_numbers,k21_sin_cos,k1_rcomp_1(np__0,k32_sin_cos))) = k1_rcomp_1(k1_real_1(np__1),np__1) ).
fof(t51_comptrig,axiom,
k2_relat_1(k2_partfun1(k1_numbers,k1_numbers,k21_sin_cos,k1_rcomp_1(k32_sin_cos,k4_real_1(np__2,k32_sin_cos)))) = k1_rcomp_1(k1_real_1(np__1),np__1) ).
fof(d1_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ( A != np__0
=> ( B = k1_comptrig(A)
<=> ( A = k2_xcmplx_0(k4_real_1(k17_complex1(A),k23_sin_cos(B)),k3_xcmplx_0(k4_real_1(k17_complex1(A),k20_sin_cos(B)),k7_complex1))
& r1_xreal_0(np__0,B)
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),B) ) ) )
& ( A = np__0
=> ( B = k1_comptrig(A)
<=> B = np__0 ) ) ) ) ) ).
fof(t52_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r1_xreal_0(np__0,k1_comptrig(A))
& ~ r1_xreal_0(k4_real_1(np__2,k32_sin_cos),k1_comptrig(A)) ) ) ).
fof(t53_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( r1_xreal_0(np__0,A)
=> k1_comptrig(k1_hahnban1(A,np__0)) = np__0 ) ) ).
fof(t54_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(np__0,A)
=> k1_comptrig(k1_hahnban1(A,np__0)) = k32_sin_cos ) ) ).
fof(t55_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_comptrig(k1_hahnban1(np__0,A)) = k6_real_1(k32_sin_cos,np__2) ) ) ).
fof(t56_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(np__0,A)
=> k1_comptrig(k1_hahnban1(np__0,A)) = k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos) ) ) ).
fof(t57_comptrig,axiom,
k1_comptrig(k2_group_1(k1_complfld)) = np__0 ).
fof(t58_comptrig,axiom,
k1_comptrig(k2_hahnban1) = k6_real_1(k32_sin_cos,np__2) ).
fof(t59_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r2_hidden(k1_comptrig(A),k2_rcomp_1(np__0,k6_real_1(k32_sin_cos,np__2)))
<=> ( ~ r1_xreal_0(k3_complex1(A),np__0)
& ~ r1_xreal_0(k4_complex1(A),np__0) ) ) ) ).
fof(t60_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r2_hidden(k1_comptrig(A),k2_rcomp_1(k6_real_1(k32_sin_cos,np__2),k32_sin_cos))
<=> ( ~ r1_xreal_0(np__0,k3_complex1(A))
& ~ r1_xreal_0(k4_complex1(A),np__0) ) ) ) ).
fof(t61_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r2_hidden(k1_comptrig(A),k2_rcomp_1(k32_sin_cos,k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos)))
<=> ( ~ r1_xreal_0(np__0,k3_complex1(A))
& ~ r1_xreal_0(np__0,k4_complex1(A)) ) ) ) ).
fof(t62_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r2_hidden(k1_comptrig(A),k2_rcomp_1(k4_real_1(k6_real_1(np__3,np__2),k32_sin_cos),k4_real_1(np__2,k32_sin_cos)))
<=> ( ~ r1_xreal_0(k3_complex1(A),np__0)
& ~ r1_xreal_0(np__0,k4_complex1(A)) ) ) ) ).
fof(t63_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ~ ( ~ r1_xreal_0(k4_complex1(A),np__0)
& r1_xreal_0(k20_sin_cos(k1_comptrig(A)),np__0) ) ) ).
fof(t64_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ~ ( ~ r1_xreal_0(np__0,k4_complex1(A))
& r1_xreal_0(np__0,k20_sin_cos(k1_comptrig(A))) ) ) ).
fof(t65_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r1_xreal_0(np__0,k4_complex1(A))
=> r1_xreal_0(np__0,k20_sin_cos(k1_comptrig(A))) ) ) ).
fof(t66_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r1_xreal_0(k4_complex1(A),np__0)
=> r1_xreal_0(k20_sin_cos(k1_comptrig(A)),np__0) ) ) ).
fof(t67_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ~ ( ~ r1_xreal_0(k3_complex1(A),np__0)
& r1_xreal_0(k23_sin_cos(k1_comptrig(A)),np__0) ) ) ).
fof(t68_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ~ ( ~ r1_xreal_0(np__0,k3_complex1(A))
& r1_xreal_0(np__0,k23_sin_cos(k1_comptrig(A))) ) ) ).
fof(t69_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r1_xreal_0(np__0,k3_complex1(A))
=> r1_xreal_0(np__0,k23_sin_cos(k1_comptrig(A))) ) ) ).
fof(t70_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ( r1_xreal_0(k3_complex1(A),np__0)
=> ( A = np__0
| r1_xreal_0(k23_sin_cos(k1_comptrig(A)),np__0) ) ) ) ).
fof(t71_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),k1_hahnban1(k23_sin_cos(A),k20_sin_cos(A)),B) = k1_hahnban1(k23_sin_cos(k4_real_1(B,A)),k20_sin_cos(k4_real_1(B,A))) ) ) ).
fof(t72_comptrig,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ ( A = k1_rlvect_1(k1_complfld)
& B = np__0 )
=> k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),A,B) = k1_hahnban1(k4_real_1(k4_power(k3_complfld(A),B),k23_sin_cos(k4_real_1(B,k1_comptrig(A)))),k4_real_1(k4_power(k3_complfld(A),B),k20_sin_cos(k4_real_1(B,k1_comptrig(A))))) ) ) ) ).
fof(t73_comptrig,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( B != np__0
=> k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),k1_hahnban1(k23_sin_cos(k6_real_1(k3_real_1(A,k4_real_1(k4_real_1(np__2,k32_sin_cos),C)),B)),k20_sin_cos(k6_real_1(k3_real_1(A,k4_real_1(k4_real_1(np__2,k32_sin_cos),C)),B))),B) = k1_hahnban1(k23_sin_cos(A),k20_sin_cos(A)) ) ) ) ) ).
fof(t74_comptrig,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( B != np__0
=> A = k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),k1_hahnban1(k4_real_1(k2_power(B,k3_complfld(A)),k23_sin_cos(k6_real_1(k3_real_1(k1_comptrig(A),k4_real_1(k4_real_1(np__2,k32_sin_cos),C)),B))),k4_real_1(k2_power(B,k3_complfld(A)),k20_sin_cos(k6_real_1(k3_real_1(k1_comptrig(A),k4_real_1(k4_real_1(np__2,k32_sin_cos),C)),B)))),B) ) ) ) ) ).
fof(d2_comptrig,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ( m1_comptrig(C,A,B)
<=> k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),C,B) = A ) ) ) ) ).
fof(t75_comptrig,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> m1_comptrig(k1_hahnban1(k4_real_1(k2_power(B,k3_complfld(A)),k23_sin_cos(k6_real_1(k3_real_1(k1_comptrig(A),k4_real_1(k4_real_1(np__2,k32_sin_cos),C)),B))),k4_real_1(k2_power(B,k3_complfld(A)),k20_sin_cos(k6_real_1(k3_real_1(k1_comptrig(A),k4_real_1(k4_real_1(np__2,k32_sin_cos),C)),B)))),A,B) ) ) ) ).
fof(t76_comptrig,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_comptrig(B,A,np__1)
=> B = A ) ) ).
fof(t77_comptrig,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_comptrig(B,k1_rlvect_1(k1_complfld),A)
=> B = k1_rlvect_1(k1_complfld) ) ) ).
fof(t78_comptrig,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_comptrig(C,B,A)
=> ( C = k1_rlvect_1(k1_complfld)
=> B = k1_rlvect_1(k1_complfld) ) ) ) ) ).
fof(s1_comptrig,axiom,
( ( ? [A] :
( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers)
& p1_s1_comptrig(A) )
& ! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ~ ( A != np__1
& p1_s1_comptrig(A)
& ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ~ ( ~ r1_xreal_0(A,B)
& p1_s1_comptrig(B) ) ) ) ) )
=> p1_s1_comptrig(np__1) ) ).
fof(dt_m1_comptrig,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k1_complfld))
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ! [C] :
( m1_comptrig(C,A,B)
=> m1_subset_1(C,u1_struct_0(k1_complfld)) ) ) ).
fof(existence_m1_comptrig,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k1_complfld))
& ~ v1_xboole_0(B)
& m1_subset_1(B,k5_numbers) )
=> ? [C] : m1_comptrig(C,A,B) ) ).
fof(dt_k1_comptrig,axiom,
! [A] :
( v1_xcmplx_0(A)
=> m1_subset_1(k1_comptrig(A),k1_numbers) ) ).
%------------------------------------------------------------------------------