SET007 Axioms: SET007+846.ax
%------------------------------------------------------------------------------
% File : SET007+846 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Groups
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : group_8 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 28 ( 0 unt; 0 def)
% Number of atoms : 335 ( 38 equ)
% Maximal formula atoms : 27 ( 11 avg)
% Number of connectives : 356 ( 49 ~; 3 |; 181 &)
% ( 6 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 11 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-3 aty)
% Number of functors : 26 ( 26 usr; 3 con; 0-3 aty)
% Number of variables : 105 ( 100 !; 5 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_group_8,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A) )
=> ~ v1_xboole_0(k14_group_2(A,B)) ) ).
fof(t1_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(B)
& k9_group_1(A) = B
& v6_group_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k7_group_1(A,C) != B ) ) ) ) ).
fof(t2_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ( ( C = E
& D = F )
=> k1_group_1(B,C,D) = k1_group_1(A,E,F) ) ) ) ) ) ) ) ).
fof(t3_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( C = D
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k6_group_1(B,E,C) = k6_group_1(A,E,D) ) ) ) ) ) ) ).
fof(t4_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( C = D
=> ! [E] :
( v1_int_1(E)
=> k6_group_1(B,E,C) = k6_group_1(A,E,D) ) ) ) ) ) ) ).
fof(t5_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( C = D
& v6_group_1(A) )
=> k7_group_1(B,C) = k7_group_1(A,D) ) ) ) ) ) ).
fof(t6_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_rlvect_1(B,C)
=> r1_tarski(k13_group_2(A,B,C),u1_struct_0(B)) ) ) ) ) ).
fof(t7_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k2_group_1(A)
& k5_group_4(A,k1_struct_0(A,B)) = k5_group_2(A) ) ) ) ).
fof(t8_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( v1_int_1(B)
=> k6_group_1(A,B,k2_group_1(A)) = k2_group_1(A) ) ) ).
fof(t9_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( v1_int_1(C)
=> k6_group_1(A,k3_xcmplx_0(C,k7_group_1(A,B)),B) = k2_group_1(A) ) ) ) ).
fof(t10_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ~ v5_group_1(B,A)
=> ! [C] :
( v1_int_1(C)
=> k6_group_1(A,C,B) = k6_group_1(A,k6_int_1(C,k7_group_1(A,B)),B) ) ) ) ) ).
fof(t11_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ~ v5_group_1(B,A)
=> v6_group_1(k5_group_4(A,k1_struct_0(A,B))) ) ) ) ).
fof(t12_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v5_group_1(B,A)
=> v5_group_1(k3_group_1(A,B),A) ) ) ) ).
fof(t13_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v5_group_1(B,A)
<=> ! [C] :
( v1_int_1(C)
=> ( k6_group_1(A,C,B) = k2_group_1(A)
=> C = np__0 ) ) ) ) ) ).
fof(t14_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ~ ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> B = k2_group_1(A) )
=> ( ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ( B = A
| r1_group_2(A,B,k5_group_2(A)) ) )
<=> ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A)
& v6_group_1(A)
& ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k9_group_1(A) = B
& v1_int_2(B) ) ) ) ) ) ).
fof(t15_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(D,k4_group_2(A,C,k3_group_2(A,B,E)))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& D = k1_group_1(A,k1_group_1(A,B,F),C)
& r2_hidden(F,E) ) ) ) ) ) ) ) ).
fof(t16_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_card_1(B) = k1_card_1(k4_group_2(A,C,k3_group_2(A,k3_group_1(A,C),B))) ) ) ) ).
fof(d1_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( D = k1_group_8(A,B,C)
<=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,D)
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& E = k10_group_2(A,C,k13_group_2(A,B,F)) ) ) ) ) ) ) ) ) ).
fof(t17_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( r2_hidden(B,k10_group_2(A,E,k13_group_2(A,D,C)))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& ? [G] :
( m1_subset_1(G,u1_struct_0(A))
& B = k1_group_1(A,k1_group_1(A,F,C),G)
& r1_rlvect_1(D,F)
& r1_rlvect_1(E,G) ) ) ) ) ) ) ) ) ).
fof(t18_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,A) )
=> ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,A) )
=> ( k10_group_2(A,E,k13_group_2(A,D,B)) = k10_group_2(A,E,k13_group_2(A,D,C))
| ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ~ ( r2_hidden(F,k10_group_2(A,E,k13_group_2(A,D,B)))
& r2_hidden(F,k10_group_2(A,E,k13_group_2(A,D,C))) ) ) ) ) ) ) ) ) ).
fof(t19_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_6(D,A,B) )
=> ( ( A = k8_group_4(A,B,C)
& r1_group_2(A,D,k9_group_2(A,B,C))
& v6_group_1(A) )
=> r1_xreal_0(k17_group_2(B,D),k17_group_2(A,C)) ) ) ) ) ) ).
fof(t20_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ~ ( v6_group_1(A)
& r1_xreal_0(k17_group_2(A,B),np__0) ) ) ) ).
fof(t21_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v6_group_1(A)
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_6(C,A,B) )
=> ! [D] :
( ( v1_group_1(D)
& m1_group_6(D,A,B) )
=> ( B = k8_group_4(B,C,D)
=> ! [E] :
( ( v1_group_1(E)
& m1_group_6(E,B,C) )
=> ( r1_group_2(B,E,k9_group_2(B,C,D))
=> ! [F] :
( ( v1_group_1(F)
& m1_group_6(F,B,D) )
=> ( r1_group_2(B,F,k9_group_2(B,C,D))
=> ! [G] :
( ( v1_group_1(G)
& m1_group_6(G,A,B) )
=> ( ( r1_group_2(B,G,k9_group_2(B,C,D))
& v1_finset_1(k14_group_2(B,D))
& v1_finset_1(k14_group_2(B,C))
& r2_int_2(k17_group_2(B,C),k17_group_2(B,D)) )
=> ( k17_group_2(B,D) = k17_group_2(C,E)
& k17_group_2(B,C) = k17_group_2(D,F) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t22_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_rlvect_1(B,C)
=> ! [D] :
( v1_int_1(D)
=> r1_rlvect_1(B,k6_group_1(A,D,C)) ) ) ) ) ) ).
fof(t23_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ~ ( A != k5_group_2(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> B = k2_group_1(A) ) ) ) ).
fof(t24_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( A = k5_group_4(A,k1_struct_0(A,B))
=> ( A = k5_group_2(A)
| ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ~ ( C != k5_group_2(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& r1_rlvect_1(C,k6_group_1(A,D,B)) ) ) ) ) ) ) ) ) ).
fof(t25_group_8,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
=> ( A != k5_group_2(A)
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ( B != k5_group_2(A)
=> ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) ) ) ) ) ) ).
fof(dt_k1_group_8,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> m1_subset_1(k1_group_8(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ).
%------------------------------------------------------------------------------