TPTP Problem File: ITP107^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP107^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Localization problem prob_542__8966104_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Localization/prob_542__8966104_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 310 ( 88 unt; 53 typ; 0 def)
% Number of atoms : 712 ( 254 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 7071 ( 57 ~; 5 |; 33 &;6619 @)
% ( 0 <=>; 357 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 93 ( 93 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 48 usr; 7 con; 0-6 aty)
% Number of variables : 976 ( 12 ^; 882 !; 15 ?; 976 :)
% ( 67 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:23:31.187
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_t_Congruence_Opartial__object_Opartial__object__ext,type,
partia1265057091ct_ext: $tType > $tType > $tType ).
thf(ty_t_Congruence_Oeq__object_Oeq__object__ext,type,
eq_eq_object_ext: $tType > $tType > $tType ).
thf(ty_t_Group_Omonoid_Omonoid__ext,type,
monoid_ext: $tType > $tType > $tType ).
thf(ty_t_Ring_Oring_Oring__ext,type,
ring_ext: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_c_AbelCoset_OA__RCOSETS,type,
a_RCOSETS:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_AbelCoset_Oa__l__coset,type,
a_l_coset:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_AbelCoset_Oa__r__coset,type,
a_r_coset:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > A > ( set @ A ) ) ).
thf(sy_c_AbelCoset_Oadditive__subgroup,type,
additive_subgroup:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_AbelCoset_Oset__add,type,
set_add:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Congruence_Oequivalence,type,
equivalence:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( eq_eq_object_ext @ A @ B ) ) > $o ) ).
thf(sy_c_Congruence_Opartial__object_Ocarrier,type,
partial_carrier:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ Z ) > ( set @ A ) ) ).
thf(sy_c_Group_OUnits,type,
units:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Group_Ogroup,type,
group:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
thf(sy_c_Group_Omonoid,type,
monoid:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > $o ) ).
thf(sy_c_Group_Omonoid_Omult,type,
mult:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A > A > A ) ).
thf(sy_c_Group_Omonoid_Oone,type,
one:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ Z ) ) > A ) ).
thf(sy_c_Group_Ounits__of,type,
units_of:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ product_unit ) ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__class__of__rng__of__frac,type,
locali1641774487f_frac:
!>[A: $tType,B: $tType,C: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ A @ B ) @ ( eq_eq_object_ext @ ( product_prod @ A @ B ) @ C ) ) > A > B > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac,type,
locali1648044335f_frac:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oeq__obj__rng__of__frac_Orec__monoid__rng__of__frac,type,
locali1768110497f_frac:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > ( partia1265057091ct_ext @ ( set @ ( product_prod @ A @ A ) ) @ ( monoid_ext @ ( set @ ( product_prod @ A @ A ) ) @ product_unit ) ) ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__crng,type,
locali7914339f_crng:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Omult__submonoid__of__rng,type,
locali1402578320of_rng:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Oset__eq__class__of__rng__of__frac,type,
locali990921413f_frac:
!>[A: $tType,B: $tType,C: $tType] : ( ( partia1265057091ct_ext @ ( product_prod @ A @ B ) @ ( eq_eq_object_ext @ ( product_prod @ A @ B ) @ C ) ) > ( set @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Localization__Mirabelle__ojffvtlkod_Osubmonoid,type,
locali1441642228monoid:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Ring_Oabelian__monoid,type,
abelian_monoid:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Ring_Oadd__pow,type,
add_pow:
!>[A: $tType,C: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ C ) ) ) > B > A > A ) ).
thf(sy_c_Ring_Ofield,type,
field:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Ring_Oring,type,
ring:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Ring_Oring_Ozero,type,
zero:
!>[A: $tType,Z: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ Z ) ) ) > A ) ).
thf(sy_c_Ring_Osemiring,type,
semiring:
!>[A: $tType,B: $tType] : ( ( partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_R,type,
r: partia1265057091ct_ext @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) ).
thf(sy_v_S,type,
s: set @ a ).
thf(sy_v_r,type,
r2: a ).
thf(sy_v_r_H,type,
r3: a ).
thf(sy_v_rel,type,
rel: partia1265057091ct_ext @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) ).
thf(sy_v_s,type,
s2: a ).
thf(sy_v_s_H,type,
s3: a ).
% Relevant facts (256)
thf(fact_0_f3,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) ) ).
% f3
thf(fact_1_f4,axiom,
( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ).
% f4
thf(fact_2_eq__obj__rng__of__frac__axioms,axiom,
locali1648044335f_frac @ a @ b @ r @ s ).
% eq_obj_rng_of_frac_axioms
thf(fact_3_f1,axiom,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r2 @ r3 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s2 @ s3 ) ) ) ).
% f1
thf(fact_4_f2,axiom,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ r3 @ r2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ s3 @ s2 ) ) ) ).
% f2
thf(fact_5_mult__submonoid__of__crng__axioms,axiom,
locali7914339f_crng @ a @ b @ r @ s ).
% mult_submonoid_of_crng_axioms
thf(fact_6_mult__submonoid__of__rng__axioms,axiom,
locali1402578320of_rng @ a @ b @ r @ s ).
% mult_submonoid_of_rng_axioms
thf(fact_7_eq__obj__rng__of__frac_Orec__monoid__rng__of__frac_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( locali1768110497f_frac @ A @ B )
= ( locali1768110497f_frac @ A @ B ) ) ).
% eq_obj_rng_of_frac.rec_monoid_rng_of_frac.cong
thf(fact_8_submonoid__axioms,axiom,
locali1441642228monoid @ a @ ( ring_ext @ a @ b ) @ r @ s ).
% submonoid_axioms
thf(fact_9_m__closed,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ s )
=> ( ( member @ a @ Y @ s )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ s ) ) ) ).
% m_closed
thf(fact_10_monoid__rng__of__frac,axiom,
monoid @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ).
% monoid_rng_of_frac
thf(fact_11_local_Osemiring__axioms,axiom,
semiring @ a @ b @ r ).
% local.semiring_axioms
thf(fact_12_one__closed,axiom,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ s ).
% one_closed
thf(fact_13_equiv__obj__rng__of__frac,axiom,
equivalence @ ( product_prod @ a @ a ) @ product_unit @ rel ).
% equiv_obj_rng_of_frac
thf(fact_14_assms_I2_J,axiom,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r3 @ s3 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% assms(2)
thf(fact_15_assms_I1_J,axiom,
member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ r2 @ s2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) ).
% assms(1)
thf(fact_16_assoc__mult__rng__of__frac,axiom,
! [R: a,S: a,T2: a,U: a,V: a,W: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ T2 @ U ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ V @ W ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ V @ W ) )
= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ V @ W ) ) ) ) ) ) ) ).
% assoc_mult_rng_of_frac
thf(fact_17_abelian__monoid__axioms,axiom,
abelian_monoid @ a @ b @ r ).
% abelian_monoid_axioms
thf(fact_18_mult__rng__of__frac__fundamental__lemma,axiom,
! [R: a,S: a,R2: a,S2: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R2 @ S2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R2 @ S2 ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ R @ R2 ) @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ S @ S2 ) ) ) ) ) ).
% mult_rng_of_frac_fundamental_lemma
thf(fact_19_local_Oring__axioms,axiom,
ring @ a @ b @ r ).
% local.ring_axioms
thf(fact_20_closed__mult__rng__of__frac,axiom,
! [R: a,S: a,T2: a,U: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ T2 @ U ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( member @ ( set @ ( product_prod @ a @ a ) ) @ ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ T2 @ U ) ) @ ( locali990921413f_frac @ a @ a @ product_unit @ rel ) ) ) ) ).
% closed_mult_rng_of_frac
thf(fact_21_submonoid_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
( ( locali1441642228monoid @ A @ B @ M @ S3 )
=> ( monoid @ A @ B @ M ) ) ).
% submonoid.axioms(1)
thf(fact_22_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( locali1648044335f_frac @ A @ B @ R3 @ S3 )
=> ( locali7914339f_crng @ A @ B @ R3 @ S3 ) ) ).
% eq_obj_rng_of_frac.axioms(2)
thf(fact_23_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( locali1402578320of_rng @ A @ B @ R3 @ S3 )
=> ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R3 @ S3 ) ) ).
% mult_submonoid_of_rng.axioms(2)
thf(fact_24_mult__submonoid__of__rng_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( locali1402578320of_rng @ A @ B @ R3 @ S3 )
=> ( ring @ A @ B @ R3 ) ) ).
% mult_submonoid_of_rng.axioms(1)
thf(fact_25_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( locali7914339f_crng @ A @ B @ R3 @ S3 )
=> ( locali1402578320of_rng @ A @ B @ R3 @ S3 ) ) ).
% mult_submonoid_of_crng.axioms(2)
thf(fact_26_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Oone__closed,axiom,
! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
( ( locali1441642228monoid @ A @ B @ M @ S3 )
=> ( member @ A @ ( one @ A @ B @ M ) @ S3 ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.one_closed
thf(fact_27_mult__submonoid__of__rng__def,axiom,
! [B: $tType,A: $tType] :
( ( locali1402578320of_rng @ A @ B )
= ( ^ [R4: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S4: set @ A] :
( ( ring @ A @ B @ R4 )
& ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R4 @ S4 ) ) ) ) ).
% mult_submonoid_of_rng_def
thf(fact_28_mult__submonoid__of__rng_Ointro,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( ring @ A @ B @ R3 )
=> ( ( locali1441642228monoid @ A @ ( ring_ext @ A @ B ) @ R3 @ S3 )
=> ( locali1402578320of_rng @ A @ B @ R3 @ S3 ) ) ) ).
% mult_submonoid_of_rng.intro
thf(fact_29_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Ois__submonoid,axiom,
! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
( ( locali1441642228monoid @ A @ B @ M @ S3 )
=> ( locali1441642228monoid @ A @ B @ M @ S3 ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.is_submonoid
thf(fact_30_eq__obj__rng__of__frac_Omonoid__rng__of__frac,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),S3: set @ A] :
( ( locali1648044335f_frac @ A @ B @ R3 @ S3 )
=> ( monoid @ ( set @ ( product_prod @ A @ A ) ) @ product_unit @ ( locali1768110497f_frac @ A @ B @ R3 @ S3 ) ) ) ).
% eq_obj_rng_of_frac.monoid_rng_of_frac
thf(fact_31_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Om__closed,axiom,
! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A,X: A,Y: A] :
( ( locali1441642228monoid @ A @ B @ M @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y @ S3 )
=> ( member @ A @ ( mult @ A @ B @ M @ X @ Y ) @ S3 ) ) ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.m_closed
thf(fact_32_right__unit__mult__rng__of__frac,axiom,
! [R: a,S: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) ) ) ).
% right_unit_mult_rng_of_frac
thf(fact_33_left__unit__mult__rng__of__frac,axiom,
! [R: a,S: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( one @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S ) ) ) ).
% left_unit_mult_rng_of_frac
thf(fact_34_zero__in__mult__submonoid,axiom,
! [R: a,S: a,R2: a,S2: a] :
( ( member @ a @ ( zero @ a @ b @ r ) @ s )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R2 @ S2 ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S )
= ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R2 @ S2 ) ) ) ) ) ).
% zero_in_mult_submonoid
thf(fact_35_non__empty__class,axiom,
! [R: a,S: a] :
( ( member @ ( product_prod @ a @ a ) @ ( product_Pair @ a @ a @ R @ S ) @ ( partial_carrier @ ( product_prod @ a @ a ) @ ( eq_eq_object_ext @ ( product_prod @ a @ a ) @ product_unit ) @ rel ) )
=> ( ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ R @ S )
!= ( bot_bot @ ( set @ ( product_prod @ a @ a ) ) ) ) ) ).
% non_empty_class
thf(fact_36_one__unique,axiom,
! [U: a] :
( ( member @ a @ U @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ! [X2: a] :
( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% one_unique
thf(fact_37_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y2 )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_38_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_39_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_40_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_41_semiring_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( abelian_monoid @ A @ B @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_42_monoid__axioms,axiom,
monoid @ a @ ( ring_ext @ a @ b ) @ r ).
% monoid_axioms
thf(fact_43_m__assoc,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ Z2 )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_44_m__comm,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_m__lcomm,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ Z2 ) )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Y @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_50_Units__closed,axiom,
! [X: a] :
( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% Units_closed
thf(fact_51_Units__l__inv__ex,axiom,
! [X: a] :
( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ? [X2: a] :
( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X2 @ X )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_52_Units__r__inv__ex,axiom,
! [X: a] :
( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ? [X2: a] :
( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ X2 )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_53_zero__closed,axiom,
member @ a @ ( zero @ a @ b @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% zero_closed
thf(fact_54_semiring__simprules_I3_J,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% semiring_simprules(3)
thf(fact_55_semiring__simprules_I4_J,axiom,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% semiring_simprules(4)
thf(fact_56_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ Y @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( member @ a @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) ).
% Units_m_closed
thf(fact_57_Units__one__closed,axiom,
member @ a @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ).
% Units_one_closed
thf(fact_58_l__null,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( zero @ a @ b @ r ) @ X )
= ( zero @ a @ b @ r ) ) ) ).
% l_null
thf(fact_59_r__null,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ) ).
% r_null
thf(fact_60_l__one,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_61_r__one,axiom,
! [X: a] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
= X ) ) ).
% r_one
thf(fact_62_Units__l__cancel,axiom,
! [X: a,Y: a,Z2: a] :
( ( member @ a @ X @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ( member @ a @ Y @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ Z2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Y )
= ( mult @ a @ ( ring_ext @ a @ b ) @ r @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ).
% Units_l_cancel
thf(fact_63_abelian__monoidE_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( abelian_monoid @ A @ B @ R3 )
=> ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_64_ring_Oring__simprules_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R3 )
=> ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_65_semiring_Osemiring__simprules_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( member @ A @ ( zero @ A @ B @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_66_abelian__monoid_Ozero__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( abelian_monoid @ A @ B @ G2 )
=> ( member @ A @ ( zero @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_67_ring_Ois__monoid,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R3 )
=> ( monoid @ A @ ( ring_ext @ A @ B ) @ R3 ) ) ).
% ring.is_monoid
thf(fact_68_semiring_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( monoid @ A @ ( ring_ext @ A @ B ) @ R3 ) ) ).
% semiring.axioms(2)
thf(fact_69_ring_Oring__simprules_I24_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
= ( zero @ A @ B @ R3 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_70_ring_Oring__simprules_I25_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
= ( zero @ A @ B @ R3 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_71_semiring_Or__null,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( zero @ A @ B @ R3 ) )
= ( zero @ A @ B @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_72_semiring_Ol__null,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( zero @ A @ B @ R3 ) @ X )
= ( zero @ A @ B @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_73_ring_Oring__simprules_I5_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_74_ring_Oring__simprules_I11_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
= ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_75_semiring_Osemiring__simprules_I3_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( member @ A @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_76_semiring_Osemiring__simprules_I8_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A,Y: A,Z2: A] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ Y ) @ Z2 )
= ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ X @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ Y @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_77_ring_Oring__simprules_I6_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R3 )
=> ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_78_semiring_Osemiring__simprules_I4_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( member @ A @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_79_ring_Oring__simprules_I12_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_80_semiring_Osemiring__simprules_I9_J,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),X: A] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( one @ A @ ( ring_ext @ A @ B ) @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_81_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_82_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_83_prod__induct7,axiom,
! [G3: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_84_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_85_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_86_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_87_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_88_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G3 ) @ E2 @ ( product_Pair @ F2 @ G3 @ F3 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_89_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_90_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_91_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_92_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_93_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_94_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_95_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X2: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_96_ring_Ois__ring,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( ring @ A @ B @ R3 )
=> ( ring @ A @ B @ R3 ) ) ).
% ring.is_ring
thf(fact_97_cring__fieldI2,axiom,
( ( ( zero @ a @ b @ r )
!= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ! [A5: a] :
( ( member @ a @ A5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( A5
!= ( zero @ a @ b @ r ) )
=> ? [X4: a] :
( ( member @ a @ X4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A5 @ X4 )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) ) )
=> ( field @ a @ b @ r ) ) ) ).
% cring_fieldI2
thf(fact_98_monoid_Or__one,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ X @ ( one @ A @ B @ G2 ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_99_monoid_Ol__one,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_100_monoid_Oone__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( monoid @ A @ B @ G2 )
=> ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ).
% monoid.one_closed
thf(fact_101_monoid_OUnits__r__inv__ex,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
& ( ( mult @ A @ B @ G2 @ X @ X2 )
= ( one @ A @ B @ G2 ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_102_monoid_OUnits__l__inv__ex,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
& ( ( mult @ A @ B @ G2 @ X2 @ X )
= ( one @ A @ B @ G2 ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_103_monoid_OUnits__inv__comm,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( ( mult @ A @ B @ G2 @ X @ Y )
= ( one @ A @ B @ G2 ) )
=> ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
=> ( ( member @ A @ Y @ ( units @ A @ B @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ Y @ X )
= ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_104_add__pow__ldistr__int,axiom,
! [A2: a,B2: a,K: int] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ int @ r @ K @ A2 ) @ B2 )
= ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_105_add__pow__rdistr__int,axiom,
! [A2: a,B2: a,K: int] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ int @ r @ K @ B2 ) )
= ( add_pow @ a @ b @ int @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_106_add_Oint__pow__closed,axiom,
! [X: a,I: int] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ int @ r @ I @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% add.int_pow_closed
thf(fact_107_add_Oint__pow__one,axiom,
! [Z2: int] :
( ( add_pow @ a @ b @ int @ r @ Z2 @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ).
% add.int_pow_one
thf(fact_108_field_Ois__ring,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( field @ A @ B @ R3 )
=> ( ring @ A @ B @ R3 ) ) ).
% field.is_ring
thf(fact_109_one__not__zero,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( field @ A @ B @ R3 )
=> ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
!= ( zero @ A @ B @ R3 ) ) ) ).
% one_not_zero
thf(fact_110_integral,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A] :
( ( field @ A @ B @ R3 )
=> ( ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 )
= ( zero @ A @ B @ R3 ) )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( A2
= ( zero @ A @ B @ R3 ) )
| ( B2
= ( zero @ A @ B @ R3 ) ) ) ) ) ) ) ).
% integral
thf(fact_111_ring_Oadd__pow__ldistr__int,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: int] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ int @ R3 @ K @ A2 ) @ B2 )
= ( add_pow @ A @ B @ int @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_112_ring_Oadd__pow__rdistr__int,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: int] :
( ( ring @ A @ B @ R3 )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ int @ R3 @ K @ B2 ) )
= ( add_pow @ A @ B @ int @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_113_monoid_Ocarrier__not__empty,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( monoid @ A @ B @ G2 )
=> ( ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% monoid.carrier_not_empty
thf(fact_114_monoid_Om__assoc,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X @ Y ) @ Z2 )
= ( mult @ A @ B @ G2 @ X @ ( mult @ A @ B @ G2 @ Y @ Z2 ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_115_monoid_Om__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( member @ A @ ( mult @ A @ B @ G2 @ X @ Y ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ) ) ).
% monoid.m_closed
thf(fact_116_monoid_OUnits__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
=> ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ) ).
% monoid.Units_closed
thf(fact_117_monoid_OUnits__m__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
=> ( ( member @ A @ Y @ ( units @ A @ B @ G2 ) )
=> ( member @ A @ ( mult @ A @ B @ G2 @ X @ Y ) @ ( units @ A @ B @ G2 ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_118_monoid_OUnits__one__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( monoid @ A @ B @ G2 )
=> ( member @ A @ ( one @ A @ B @ G2 ) @ ( units @ A @ B @ G2 ) ) ) ).
% monoid.Units_one_closed
thf(fact_119_monoidI,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ! [X2: A,Y3: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
=> ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ! [X2: A,Y3: A,Z3: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
= ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
= X2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ X2 @ ( one @ A @ B @ G2 ) )
= X2 ) )
=> ( monoid @ A @ B @ G2 ) ) ) ) ) ) ).
% monoidI
thf(fact_120_Group_Omonoid__def,axiom,
! [B: $tType,A: $tType] :
( ( monoid @ A @ B )
= ( ^ [G5: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ! [X3: A,Y4: A] :
( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( member @ A @ ( mult @ A @ B @ G5 @ X3 @ Y4 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) ) ) )
& ! [X3: A,Y4: A,Z4: A] :
( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( ( member @ A @ Y4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( ( member @ A @ Z4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( ( mult @ A @ B @ G5 @ ( mult @ A @ B @ G5 @ X3 @ Y4 ) @ Z4 )
= ( mult @ A @ B @ G5 @ X3 @ ( mult @ A @ B @ G5 @ Y4 @ Z4 ) ) ) ) ) )
& ( member @ A @ ( one @ A @ B @ G5 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( ( mult @ A @ B @ G5 @ ( one @ A @ B @ G5 ) @ X3 )
= X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G5 ) )
=> ( ( mult @ A @ B @ G5 @ X3 @ ( one @ A @ B @ G5 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_121_Group_Omonoid_Ointro,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ! [X2: A,Y3: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
=> ( ! [X2: A,Y3: A,Z3: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
= ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
=> ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
= X2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ X2 @ ( one @ A @ B @ G2 ) )
= X2 ) )
=> ( monoid @ A @ B @ G2 ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_122_monoid_Oinv__unique,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),Y: A,X: A,Y2: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( ( mult @ A @ B @ G2 @ Y @ X )
= ( one @ A @ B @ G2 ) )
=> ( ( ( mult @ A @ B @ G2 @ X @ Y2 )
= ( one @ A @ B @ G2 ) )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_123_monoid_Oone__unique,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),U: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ U @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ U @ X2 )
= X2 ) )
=> ( U
= ( one @ A @ B @ G2 ) ) ) ) ) ).
% monoid.one_unique
thf(fact_124_monoid_OUnits__l__cancel,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
( ( monoid @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( units @ A @ B @ G2 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( ( mult @ A @ B @ G2 @ X @ Y )
= ( mult @ A @ B @ G2 @ X @ Z2 ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_125_subset,axiom,
ord_less_eq @ ( set @ a ) @ s @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ).
% subset
thf(fact_126_add__pow__rdistr,axiom,
! [A2: a,B2: a,K: nat] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ ( add_pow @ a @ b @ nat @ r @ K @ B2 ) )
= ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_rdistr
thf(fact_127_add__pow__ldistr,axiom,
! [A2: a,B2: a,K: nat] :
( ( member @ a @ A2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ B2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ ( add_pow @ a @ b @ nat @ r @ K @ A2 ) @ B2 )
= ( add_pow @ a @ b @ nat @ r @ K @ ( mult @ a @ ( ring_ext @ a @ b ) @ r @ A2 @ B2 ) ) ) ) ) ).
% add_pow_ldistr
thf(fact_128_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_129_group__l__invI,axiom,
( ! [X2: a] :
( ( member @ a @ X2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ? [Xa: a] :
( ( member @ a @ Xa @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
& ( ( mult @ a @ ( ring_ext @ a @ b ) @ r @ Xa @ X2 )
= ( one @ a @ ( ring_ext @ a @ b ) @ r ) ) ) )
=> ( group @ a @ ( ring_ext @ a @ b ) @ r ) ) ).
% group_l_invI
thf(fact_130_carrier__not__empty,axiom,
( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
!= ( bot_bot @ ( set @ a ) ) ) ).
% carrier_not_empty
thf(fact_131_add_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ a @ ( add_pow @ a @ b @ nat @ r @ N @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ).
% add.nat_pow_closed
thf(fact_132_add_Onat__pow__one,axiom,
! [N: nat] :
( ( add_pow @ a @ b @ nat @ r @ N @ ( zero @ a @ b @ r ) )
= ( zero @ a @ b @ r ) ) ).
% add.nat_pow_one
thf(fact_133_group_Ois__group,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( group @ A @ B @ G2 )
=> ( group @ A @ B @ G2 ) ) ).
% group.is_group
thf(fact_134_group_OUnits,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( group @ A @ B @ G2 )
=> ( ord_less_eq @ ( set @ A ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) @ ( units @ A @ B @ G2 ) ) ) ).
% group.Units
thf(fact_135_group_Ois__monoid,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( group @ A @ B @ G2 )
=> ( monoid @ A @ B @ G2 ) ) ).
% group.is_monoid
thf(fact_136_Group_Ogroup_Oright__cancel,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A,Z2: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Z2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( ( mult @ A @ B @ G2 @ Y @ X )
= ( mult @ A @ B @ G2 @ Z2 @ X ) )
= ( Y = Z2 ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_137_group_OUnits__eq,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( group @ A @ B @ G2 )
=> ( ( units @ A @ B @ G2 )
= ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) ).
% group.Units_eq
thf(fact_138_Localization__Mirabelle__ojffvtlkod_Osubmonoid_Osubset,axiom,
! [B: $tType,A: $tType,M: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),S3: set @ A] :
( ( locali1441642228monoid @ A @ B @ M @ S3 )
=> ( ord_less_eq @ ( set @ A ) @ S3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ M ) ) ) ).
% Localization_Mirabelle_ojffvtlkod.submonoid.subset
thf(fact_139_group_Or__cancel__one_H,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( X
= ( mult @ A @ B @ G2 @ A2 @ X ) )
= ( A2
= ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_140_group_Ol__cancel__one_H,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( X
= ( mult @ A @ B @ G2 @ X @ A2 ) )
= ( A2
= ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_141_group_Or__cancel__one,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( ( mult @ A @ B @ G2 @ A2 @ X )
= X )
= ( A2
= ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_142_group_Ol__cancel__one,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,A2: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( ( mult @ A @ B @ G2 @ X @ A2 )
= X )
= ( A2
= ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_143_group_Or__inv__ex,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
& ( ( mult @ A @ B @ G2 @ X @ X2 )
= ( one @ A @ B @ G2 ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_144_group_Ol__inv__ex,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A] :
( ( group @ A @ B @ G2 )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
& ( ( mult @ A @ B @ G2 @ X2 @ X )
= ( one @ A @ B @ G2 ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_145_group_Oinv__comm,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B ),X: A,Y: A] :
( ( group @ A @ B @ G2 )
=> ( ( ( mult @ A @ B @ G2 @ X @ Y )
= ( one @ A @ B @ G2 ) )
=> ( ( member @ A @ X @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( member @ A @ Y @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ Y @ X )
= ( one @ A @ B @ G2 ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_146_groupI,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( member @ A @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) ) ) )
=> ( ( member @ A @ ( one @ A @ B @ G2 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ! [Z3: A] :
( ( member @ A @ Z3 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( mult @ A @ B @ G2 @ X2 @ Y3 ) @ Z3 )
= ( mult @ A @ B @ G2 @ X2 @ ( mult @ A @ B @ G2 @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ( ( mult @ A @ B @ G2 @ ( one @ A @ B @ G2 ) @ X2 )
= X2 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ? [Xa: A] :
( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
& ( ( mult @ A @ B @ G2 @ Xa @ X2 )
= ( one @ A @ B @ G2 ) ) ) )
=> ( group @ A @ B @ G2 ) ) ) ) ) ) ).
% groupI
thf(fact_147_monoid_Ogroup__l__invI,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( monoid @ A @ B @ G2 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
=> ? [Xa: A] :
( ( member @ A @ Xa @ ( partial_carrier @ A @ ( monoid_ext @ A @ B ) @ G2 ) )
& ( ( mult @ A @ B @ G2 @ Xa @ X2 )
= ( one @ A @ B @ G2 ) ) ) )
=> ( group @ A @ B @ G2 ) ) ) ).
% monoid.group_l_invI
thf(fact_148_semiring_Oadd__pow__rdistr,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ ( add_pow @ A @ B @ nat @ R3 @ K @ B2 ) )
= ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_149_semiring_Oadd__pow__ldistr,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A2: A,B2: A,K: nat] :
( ( semiring @ A @ B @ R3 )
=> ( ( member @ A @ A2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( member @ A @ B2 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) )
=> ( ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ ( add_pow @ A @ B @ nat @ R3 @ K @ A2 ) @ B2 )
= ( add_pow @ A @ B @ nat @ R3 @ K @ ( mult @ A @ ( ring_ext @ A @ B ) @ R3 @ A2 @ B2 ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_150_a__lcos__mult__one,axiom,
! [M: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ M @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_l_coset @ a @ b @ r @ ( zero @ a @ b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_151_set__add__closed,axiom,
! [A4: set @ a,B5: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ A4 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ B5 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ A4 @ B5 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% set_add_closed
thf(fact_152_setadd__subset__G,axiom,
! [H: set @ a,K2: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( set_add @ a @ b @ r @ H @ K2 ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_153_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_154_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_155_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_156_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_157_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_158_subsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% subsetI
thf(fact_159_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_160_a__l__coset__subset__G,axiom,
! [H: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_l_coset @ a @ b @ r @ X @ H ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_161_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_162_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_163_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_164_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_165_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_166_in__mono,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_167_subsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_168_equalityE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_169_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_170_equalityD1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_171_equalityD2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_172_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_173_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_174_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_175_subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_176_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5 )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_177_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_178_units__group,axiom,
group @ a @ product_unit @ ( units_of @ a @ ( ring_ext @ a @ b ) @ r ) ).
% units_group
thf(fact_179_add__additive__subgroups,axiom,
! [H: set @ a,K2: set @ a] :
( ( additive_subgroup @ a @ b @ H @ r )
=> ( ( additive_subgroup @ a @ b @ K2 @ r )
=> ( additive_subgroup @ a @ b @ ( set_add @ a @ b @ r @ H @ K2 ) @ r ) ) ) ).
% add_additive_subgroups
thf(fact_180_a__rcos__assoc__lcos,axiom,
! [H: set @ a,K2: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( set_add @ a @ b @ r @ ( a_r_coset @ a @ b @ r @ H @ X ) @ K2 )
= ( set_add @ a @ b @ r @ H @ ( a_l_coset @ a @ b @ r @ X @ K2 ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_181_a__r__coset__subset__G,axiom,
! [H: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ord_less_eq @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ H @ X ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_182_a__setmult__rcos__assoc,axiom,
! [H: set @ a,K2: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( ord_less_eq @ ( set @ a ) @ K2 @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( set_add @ a @ b @ r @ H @ ( a_r_coset @ a @ b @ r @ K2 @ X ) )
= ( a_r_coset @ a @ b @ r @ ( set_add @ a @ b @ r @ H @ K2 ) @ X ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_183_units__of__units,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( units @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
= ( units @ A @ B @ G2 ) ) ).
% units_of_units
thf(fact_184_a__coset__add__zero,axiom,
! [M: set @ a] :
( ( ord_less_eq @ ( set @ a ) @ M @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( a_r_coset @ a @ b @ r @ M @ ( zero @ a @ b @ r ) )
= M ) ) ).
% a_coset_add_zero
thf(fact_185_units__of__carrier,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( partial_carrier @ A @ ( monoid_ext @ A @ product_unit ) @ ( units_of @ A @ B @ G2 ) )
= ( units @ A @ B @ G2 ) ) ).
% units_of_carrier
thf(fact_186_monoid_Ounits__group,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( monoid @ A @ B @ G2 )
=> ( group @ A @ product_unit @ ( units_of @ A @ B @ G2 ) ) ) ).
% monoid.units_group
thf(fact_187_units__of__mult,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( mult @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
= ( mult @ A @ B @ G2 ) ) ).
% units_of_mult
thf(fact_188_units__of__one,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ B )] :
( ( one @ A @ product_unit @ ( units_of @ A @ B @ G2 ) )
= ( one @ A @ B @ G2 ) ) ).
% units_of_one
thf(fact_189_a__rcosetsI,axiom,
! [H: set @ a,X: a] :
( ( ord_less_eq @ ( set @ a ) @ H @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( ( member @ a @ X @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) )
=> ( member @ ( set @ a ) @ ( a_r_coset @ a @ b @ r @ H @ X ) @ ( a_RCOSETS @ a @ b @ r @ H ) ) ) ) ).
% a_rcosetsI
thf(fact_190_one__zeroI,axiom,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
= ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
=> ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
= ( zero @ a @ b @ r ) ) ) ).
% one_zeroI
thf(fact_191_one__zeroD,axiom,
( ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
= ( zero @ a @ b @ r ) )
=> ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
= ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) ) ).
% one_zeroD
thf(fact_192_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_193_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_194_insertCI,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B5 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_195_carrier__one__not__zero,axiom,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
!= ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
= ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
!= ( zero @ a @ b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_196_carrier__one__zero,axiom,
( ( ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r )
= ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) )
= ( ( one @ a @ ( ring_ext @ a @ b ) @ r )
= ( zero @ a @ b @ r ) ) ) ).
% carrier_one_zero
thf(fact_197_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_198_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% insert_subset
thf(fact_199_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_200_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_201_subset__singleton__iff,axiom,
! [A: $tType,X5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ( X5
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_202_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_203_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B7: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_204_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_205_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ B5 ) )
= ( ( ( A2 = B2 )
=> ( A4 = B5 ) )
& ( ( A2 != B2 )
=> ? [C5: set @ A] :
( ( A4
= ( insert @ A @ B2 @ C5 ) )
& ~ ( member @ A @ B2 @ C5 )
& ( B5
= ( insert @ A @ A2 @ C5 ) )
& ~ ( member @ A @ A2 @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_206_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_207_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B5 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B5 ) )
= ( A4 = B5 ) ) ) ) ).
% insert_ident
thf(fact_208_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B7: set @ A] :
( ( A4
= ( insert @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_209_insertI2,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( member @ A @ A2 @ B5 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_210_insertI1,axiom,
! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% insertI1
thf(fact_211_insertE,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_212_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_213_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_214_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_215_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_216_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_217_insert__mono,axiom,
! [A: $tType,C4: set @ A,D4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% insert_mono
thf(fact_218_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% subset_insert
thf(fact_219_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_220_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_221_semiring_Ocarrier__one__not__zero,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
!= ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
!= ( zero @ A @ B @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_222_semiring_Ocarrier__one__zero,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
= ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
= ( zero @ A @ B @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_223_semiring_Oone__zeroI,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
= ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
= ( zero @ A @ B @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_224_semiring_Oone__zeroD,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( semiring @ A @ B @ R3 )
=> ( ( ( one @ A @ ( ring_ext @ A @ B ) @ R3 )
= ( zero @ A @ B @ R3 ) )
=> ( ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 )
= ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% semiring.one_zeroD
thf(fact_225_field__intro2,axiom,
( ( ( zero @ a @ b @ r )
!= ( one @ a @ ( ring_ext @ a @ b ) @ r ) )
=> ( ! [X2: a] :
( ( member @ a @ X2 @ ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( member @ a @ X2 @ ( units @ a @ ( ring_ext @ a @ b ) @ r ) ) )
=> ( field @ a @ b @ r ) ) ) ).
% field_intro2
thf(fact_226_cring__fieldI,axiom,
( ( ( units @ a @ ( ring_ext @ a @ b ) @ r )
= ( minus_minus @ ( set @ a ) @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) @ ( insert @ a @ ( zero @ a @ b @ r ) @ ( bot_bot @ ( set @ a ) ) ) ) )
=> ( field @ a @ b @ r ) ) ).
% cring_fieldI
thf(fact_227_DiffI,axiom,
! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( ~ ( member @ A @ C3 @ B5 )
=> ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% DiffI
thf(fact_228_Diff__iff,axiom,
! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
= ( ( member @ A @ C3 @ A4 )
& ~ ( member @ A @ C3 @ B5 ) ) ) ).
% Diff_iff
thf(fact_229_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ).
% Diff_idemp
thf(fact_230_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_231_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_232_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_233_insert__Diff1,axiom,
! [A: $tType,X: A,B5: set @ A,A4: set @ A] :
( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_234_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_235_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_236_insert__Diff__single,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A4 ) ) ).
% insert_Diff_single
thf(fact_237_insert__Diff__if,axiom,
! [A: $tType,X: A,B5: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B5 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_238_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_239_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_240_insert__Diff,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_241_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A2: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_242_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B5 @ C4 ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_243_double__diff,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_244_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ A4 ) ).
% Diff_subset
thf(fact_245_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C4: set @ A,D4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_246_DiffE,axiom,
! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
=> ~ ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% DiffE
thf(fact_247_DiffD1,axiom,
! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
=> ( member @ A @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_248_DiffD2,axiom,
! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A4 @ B5 ) )
=> ~ ( member @ A @ C3 @ B5 ) ) ).
% DiffD2
thf(fact_249_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_250_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_251_field_Ofield__Units,axiom,
! [B: $tType,A: $tType,R3: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) )] :
( ( field @ A @ B @ R3 )
=> ( ( units @ A @ ( ring_ext @ A @ B ) @ R3 )
= ( minus_minus @ ( set @ A ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ R3 ) @ ( insert @ A @ ( zero @ A @ B @ R3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% field.field_Units
thf(fact_252_rcosets__subset__PowG,axiom,
! [H: set @ a] :
( ( additive_subgroup @ a @ b @ H @ r )
=> ( ord_less_eq @ ( set @ ( set @ a ) ) @ ( a_RCOSETS @ a @ b @ r @ H ) @ ( pow @ a @ ( partial_carrier @ a @ ( monoid_ext @ a @ ( ring_ext @ a @ b ) ) @ r ) ) ) ) ).
% rcosets_subset_PowG
thf(fact_253_abelian__monoid_Oset__add__closed,axiom,
! [B: $tType,A: $tType,G2: partia1265057091ct_ext @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ),A4: set @ A,B5: set @ A] :
( ( abelian_monoid @ A @ B @ G2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( set_add @ A @ B @ G2 @ A4 @ B5 ) @ ( partial_carrier @ A @ ( monoid_ext @ A @ ( ring_ext @ A @ B ) ) @ G2 ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_254_Pow__singleton__iff,axiom,
! [A: $tType,X5: set @ A,Y6: set @ A] :
( ( ( pow @ A @ X5 )
= ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X5
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_255_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
% Conjectures (1)
thf(conj_0,conjecture,
( ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) )
= ( mult @ ( set @ ( product_prod @ a @ a ) ) @ product_unit @ ( locali1768110497f_frac @ a @ b @ r @ s ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r3 @ s3 ) @ ( locali1641774487f_frac @ a @ a @ product_unit @ rel @ r2 @ s2 ) ) ) ).
%------------------------------------------------------------------------------