SET007 Axioms: SET007+338.ax
%------------------------------------------------------------------------------
% File : SET007+338 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Isomorphisms of Cyclic Groups. Some Properties of Cyclic 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 : gr_cy_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 6 unt; 0 def)
% Number of atoms : 390 ( 49 equ)
% Maximal formula atoms : 31 ( 10 avg)
% Number of connectives : 412 ( 58 ~; 2 |; 236 &)
% ( 4 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 21 ( 21 usr; 6 con; 0-3 aty)
% Number of variables : 95 ( 88 !; 7 ?)
% 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_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k10_group_5(A))
& v1_group_1(k10_group_5(A))
& v2_group_1(k10_group_5(A))
& v3_group_1(k10_group_5(A))
& v4_group_1(k10_group_5(A))
& v1_group_3(k10_group_5(A),A) ) ) ).
fof(t1_gr_cy_2,axiom,
$true ).
fof(t2_gr_cy_2,axiom,
$true ).
fof(t3_gr_cy_2,axiom,
$true ).
fof(t4_gr_cy_2,axiom,
$true ).
fof(t5_gr_cy_2,axiom,
$true ).
fof(t6_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(k7_group_1(A,B),np__1)
& B = k6_group_1(A,D,C)
& D = np__0 ) ) ) ) ) ).
fof(t7_gr_cy_2,axiom,
$true ).
fof(t8_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_rlvect_1(k5_group_4(A,k1_struct_0(A,B)),B) ) ) ).
fof(t9_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( C = D
=> k5_group_4(A,k1_struct_0(A,C)) = k5_group_4(B,k1_struct_0(B,D)) ) ) ) ) ) ).
fof(t10_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ~ v3_struct_0(k5_group_4(A,k1_struct_0(A,B)))
& v3_group_1(k5_group_4(A,k1_struct_0(A,B)))
& v4_group_1(k5_group_4(A,k1_struct_0(A,B)))
& v1_gr_cy_1(k5_group_4(A,k1_struct_0(A,B)))
& l1_group_1(k5_group_4(A,k1_struct_0(A,B))) ) ) ) ).
fof(t11_gr_cy_2,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_int_1(D)
& C = k6_group_1(A,D,B) ) )
<=> A = k5_group_4(A,k1_struct_0(A,B)) ) ) ) ).
fof(t12_gr_cy_2,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))
=> ( v6_group_1(A)
=> ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& C = k6_group_1(A,D,B) ) )
<=> A = k5_group_4(A,k1_struct_0(A,B)) ) ) ) ) ).
fof(t13_gr_cy_2,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))
=> ( ( v6_group_1(A)
& A = k5_group_4(A,k1_struct_0(A,B)) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r1_group_2(A,C,k5_group_4(A,k1_struct_0(A,k6_group_1(A,D,B)))) ) ) ) ) ) ).
fof(t14_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( v6_group_1(A)
& A = k5_group_4(A,k1_struct_0(A,B))
& k9_group_1(A) = C
& C = k2_nat_1(D,E) )
=> k7_group_1(A,k6_group_1(A,D,B)) = E ) ) ) ) ) ) ).
fof(t15_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_nat_1(C,D)
=> r1_rlvect_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B))),k6_group_1(A,D,B)) ) ) ) ) ) ).
fof(t16_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( v6_group_1(A)
& k9_group_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B)))) = k9_group_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,D,B))))
& r1_rlvect_1(k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B))),k6_group_1(A,D,B)) )
=> r1_group_2(A,k5_group_4(A,k1_struct_0(A,k6_group_1(A,C,B))),k5_group_4(A,k1_struct_0(A,k6_group_1(A,D,B)))) ) ) ) ) ) ).
fof(t17_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( v6_group_1(A)
& k9_group_1(A) = D
& A = k5_group_4(A,k1_struct_0(A,C))
& k9_group_1(B) = E
& B = k5_group_4(A,k1_struct_0(A,k6_group_1(A,F,C))) )
=> r1_nat_1(D,k2_nat_1(F,E)) ) ) ) ) ) ) ) ).
fof(t18_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_group_1(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ( ( v6_group_1(C)
& C = k5_group_4(C,k1_struct_0(C,D))
& k9_group_1(C) = A )
=> ( C = k5_group_4(C,k1_struct_0(C,k6_group_1(C,B,D)))
<=> k6_nat_1(B,A) = np__1 ) ) ) ) ) ) ).
fof(t19_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( A = k5_group_4(A,k1_struct_0(A,C))
& r1_rlvect_1(B,C) )
=> g1_group_1(u1_struct_0(A),u1_group_1(A)) = g1_group_1(u1_struct_0(B),u1_group_1(B)) ) ) ) ) ).
fof(t20_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_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))
=> ( v6_group_1(A)
<=> ? [C] :
( v1_int_1(C)
& ? [D] :
( v1_int_1(D)
& C != D
& k6_group_1(A,C,B) = k6_group_1(A,D,B) ) ) ) ) ) ) ).
fof(d1_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k5_gr_cy_1(A)))
=> k1_gr_cy_2(A,B) = B ) ) ) ).
fof(t21_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) )
=> ( ( v6_group_1(B)
& k9_group_1(B) = A )
=> r2_group_6(k5_gr_cy_1(A),B) ) ) ) ).
fof(t22_gr_cy_2,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) )
=> ( ~ v6_group_1(A)
=> r2_group_6(k3_gr_cy_1,A) ) ) ).
fof(t23_gr_cy_2,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) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) )
=> ( ( v6_group_1(B)
& v6_group_1(A)
& k9_group_1(B) = k9_group_1(A) )
=> r2_group_6(B,A) ) ) ) ).
fof(t24_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v1_group_1(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ( ( v6_group_1(B)
& v6_group_1(C)
& k9_group_1(B) = A
& k9_group_1(C) = A
& v1_int_2(A) )
=> r2_group_6(B,C) ) ) ) ) ).
fof(t25_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ( ( v6_group_1(A)
& v6_group_1(B)
& k9_group_1(A) = np__2
& k9_group_1(B) = np__2 )
=> r2_group_6(A,B) ) ) ) ).
fof(t26_gr_cy_2,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)
& k9_group_1(A) = np__2 )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ( r1_group_2(A,B,k5_group_2(A))
| B = A ) ) ) ) ).
fof(t27_gr_cy_2,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)
& k9_group_1(A) = np__2 )
=> ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) ) ) ) ).
fof(t28_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ( ( v6_group_1(B)
& ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B)
& k9_group_1(B) = A )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r1_nat_1(C,A)
& ! [D] :
( ( v1_group_1(D)
& m1_group_2(D,B) )
=> ~ ( k9_group_1(D) = C
& ! [E] :
( ( v1_group_1(E)
& m1_group_2(E,B) )
=> ( k9_group_1(E) = C
=> r1_group_2(B,E,D) ) ) ) ) ) ) ) ) ) ).
fof(t29_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_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))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& l1_group_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(C),u1_struct_0(A))
& v1_group_6(D,C,A)
& m2_relset_1(D,u1_struct_0(C),u1_struct_0(A)) )
=> ( r1_rlvect_1(k13_group_6(C,A,D),B)
=> v3_group_6(D,C,A) ) ) ) ) ) ) ).
fof(t30_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) )
=> ~ ( v6_group_1(B)
& k9_group_1(B) = A
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k2_nat_1(np__2,C) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ~ ( k7_group_1(B,C) = np__2
& ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( k7_group_1(B,D) = np__2
=> C = D ) ) ) ) ) ) ) ).
fof(t31_gr_cy_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) )
=> ~ ( v6_group_1(B)
& k9_group_1(B) = A
& ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k2_nat_1(np__2,C) )
& ! [C] :
( m1_group_2(C,B)
=> ~ ( k9_group_1(C) = np__2
& ~ v3_struct_0(C)
& v3_group_1(C)
& v4_group_1(C)
& v1_gr_cy_1(C)
& l1_group_1(C) ) ) ) ) ) ).
fof(t32_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(A))
& v1_group_6(C,B,A)
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(A)) )
=> ( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) )
=> ( ~ v3_struct_0(k13_group_6(B,A,C))
& v3_group_1(k13_group_6(B,A,C))
& v4_group_1(k13_group_6(B,A,C))
& v1_gr_cy_1(k13_group_6(B,A,C))
& l1_group_1(k13_group_6(B,A,C)) ) ) ) ) ) ).
fof(t33_gr_cy_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ( r2_group_6(A,B)
=> ( ( ~ ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(A) )
& ~ ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_gr_cy_1(B)
& l1_group_1(B) ) )
| ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& v1_gr_cy_1(A)
& l1_group_1(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_gr_cy_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k5_gr_cy_1(A))) )
=> m2_subset_1(k1_gr_cy_2(A,B),k1_numbers,k5_numbers) ) ).
%------------------------------------------------------------------------------