SET007 Axioms: SET007+274.ax
%------------------------------------------------------------------------------
% File : SET007+274 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Ternary Fields
% 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 : algstr_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 37 ( 10 unt; 0 def)
% Number of atoms : 207 ( 52 equ)
% Maximal formula atoms : 51 ( 5 avg)
% Number of connectives : 194 ( 24 ~; 3 |; 69 &)
% ( 2 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 18 ( 18 usr; 5 con; 0-8 aty)
% Number of variables : 110 ( 100 !; 10 ?)
% 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(rc1_algstr_3,axiom,
? [A] :
( l1_algstr_3(A)
& v1_algstr_3(A) ) ).
fof(rc2_algstr_3,axiom,
? [A] :
( l1_algstr_3(A)
& ~ v3_struct_0(A) ) ).
fof(fc1_algstr_3,axiom,
( ~ v3_struct_0(k4_algstr_3)
& v1_algstr_3(k4_algstr_3) ) ).
fof(rc3_algstr_3,axiom,
? [A] :
( l1_algstr_3(A)
& ~ v3_struct_0(A)
& v1_algstr_3(A)
& v2_algstr_3(A) ) ).
fof(d1_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_algstr_3(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))
=> k1_algstr_3(A,B,C,D) = k2_multop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u2_algstr_3(A),B,C,D) ) ) ) ) ).
fof(d2_algstr_3,axiom,
$true ).
fof(d3_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_algstr_3(A) )
=> k2_algstr_3(A) = u1_algstr_3(A) ) ).
fof(d4_algstr_3,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers)
& m2_relset_1(A,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers) )
=> ( A = k3_algstr_3
<=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k2_multop_1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,A,B,C,D) = k3_real_1(k4_real_1(B,C),D) ) ) ) ) ) ).
fof(d5_algstr_3,axiom,
k4_algstr_3 = g1_algstr_3(k1_numbers,np__0,np__1,k3_algstr_3) ).
fof(d6_algstr_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k4_algstr_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_algstr_3))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_algstr_3))
=> k5_algstr_3(A,B,C) = k2_multop_1(u1_struct_0(k4_algstr_3),u1_struct_0(k4_algstr_3),u1_struct_0(k4_algstr_3),u1_struct_0(k4_algstr_3),u2_algstr_3(k4_algstr_3),A,B,C) ) ) ) ).
fof(t1_algstr_3,axiom,
$true ).
fof(t2_algstr_3,axiom,
$true ).
fof(t3_algstr_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( A != B
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> k3_real_1(k4_real_1(A,E),C) != k3_real_1(k4_real_1(B,E),D) ) ) ) ) ) ) ).
fof(t4_algstr_3,axiom,
$true ).
fof(t5_algstr_3,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k4_algstr_3))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_algstr_3))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_algstr_3))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( A = D
& B = E
& C = F )
=> k1_algstr_3(k4_algstr_3,A,B,C) = k3_real_1(k4_real_1(D,E),F) ) ) ) ) ) ) ) ).
fof(t6_algstr_3,axiom,
np__0 = k1_rlvect_1(k4_algstr_3) ).
fof(t7_algstr_3,axiom,
np__1 = k2_algstr_3(k4_algstr_3) ).
fof(d7_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_algstr_3(A) )
=> ( v2_algstr_3(A)
<=> ( k1_rlvect_1(A) != k2_algstr_3(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_algstr_3(A,B,k2_algstr_3(A),k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_algstr_3(A,k2_algstr_3(A),B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_algstr_3(A,B,k1_rlvect_1(A),C) = C ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_algstr_3(A,k1_rlvect_1(A),B,C) = C ) )
& ! [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,u1_struct_0(A))
& k1_algstr_3(A,B,C,E) = D ) ) ) )
& ! [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,u1_struct_0(A))
=> ( k1_algstr_3(A,B,C,D) = k1_algstr_3(A,B,C,E)
=> D = E ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( B != C
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& ? [G] :
( m1_subset_1(G,u1_struct_0(A))
& k1_algstr_3(A,F,B,G) = D
& k1_algstr_3(A,F,C,G) = E ) ) ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( B != C
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& k1_algstr_3(A,B,F,D) = k1_algstr_3(A,C,F,E) ) ) ) ) ) )
& ! [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,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ~ ( k1_algstr_3(A,D,B,F) = k1_algstr_3(A,E,B,G)
& k1_algstr_3(A,D,C,F) = k1_algstr_3(A,E,C,G)
& B != C
& D != E ) ) ) ) ) ) ) ) ) ) ).
fof(t8_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_3(A)
& l1_algstr_3(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,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( k1_algstr_3(A,D,B,E) = k1_algstr_3(A,F,B,G)
& k1_algstr_3(A,D,C,E) = k1_algstr_3(A,F,C,G) )
=> ( B = C
| ( D = F
& E = G ) ) ) ) ) ) ) ) ) ) ).
fof(t9_algstr_3,axiom,
$true ).
fof(t10_algstr_3,axiom,
$true ).
fof(t11_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_3(A)
& l1_algstr_3(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k1_algstr_3(A,B,E,C) = D ) ) ) ) ) ) ).
fof(t12_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_3(A)
& l1_algstr_3(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,u1_struct_0(A))
=> ( k1_algstr_3(A,B,C,D) = k1_algstr_3(A,B,E,D)
=> ( B = k1_rlvect_1(A)
| C = E ) ) ) ) ) ) ) ).
fof(t13_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_3(A)
& l1_algstr_3(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k1_algstr_3(A,E,B,C) = D ) ) ) ) ) ) ).
fof(t14_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_algstr_3(A)
& l1_algstr_3(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,u1_struct_0(A))
=> ( k1_algstr_3(A,C,B,D) = k1_algstr_3(A,E,B,D)
=> ( B = k1_rlvect_1(A)
| C = E ) ) ) ) ) ) ) ).
fof(dt_l1_algstr_3,axiom,
! [A] :
( l1_algstr_3(A)
=> l2_struct_0(A) ) ).
fof(existence_l1_algstr_3,axiom,
? [A] : l1_algstr_3(A) ).
fof(abstractness_v1_algstr_3,axiom,
! [A] :
( l1_algstr_3(A)
=> ( v1_algstr_3(A)
=> A = g1_algstr_3(u1_struct_0(A),u2_struct_0(A),u1_algstr_3(A),u2_algstr_3(A)) ) ) ).
fof(dt_k1_algstr_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_algstr_3(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(A)) )
=> m1_subset_1(k1_algstr_3(A,B,C,D),u1_struct_0(A)) ) ).
fof(dt_k2_algstr_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_algstr_3(A) )
=> m1_subset_1(k2_algstr_3(A),u1_struct_0(A)) ) ).
fof(dt_k3_algstr_3,axiom,
( v1_funct_1(k3_algstr_3)
& v1_funct_2(k3_algstr_3,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers)
& m2_relset_1(k3_algstr_3,k3_zfmisc_1(k1_numbers,k1_numbers,k1_numbers),k1_numbers) ) ).
fof(dt_k4_algstr_3,axiom,
( v1_algstr_3(k4_algstr_3)
& l1_algstr_3(k4_algstr_3) ) ).
fof(dt_k5_algstr_3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,u1_struct_0(k4_algstr_3))
& m1_subset_1(B,u1_struct_0(k4_algstr_3))
& m1_subset_1(C,u1_struct_0(k4_algstr_3)) )
=> m1_subset_1(k5_algstr_3(A,B,C),u1_struct_0(k4_algstr_3)) ) ).
fof(dt_u1_algstr_3,axiom,
! [A] :
( l1_algstr_3(A)
=> m1_subset_1(u1_algstr_3(A),u1_struct_0(A)) ) ).
fof(dt_u2_algstr_3,axiom,
! [A] :
( l1_algstr_3(A)
=> ( v1_funct_1(u2_algstr_3(A))
& v1_funct_2(u2_algstr_3(A),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u2_algstr_3(A),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_g1_algstr_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(B,A)
& m1_subset_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,k3_zfmisc_1(A,A,A),A)
& m1_relset_1(D,k3_zfmisc_1(A,A,A),A) )
=> ( v1_algstr_3(g1_algstr_3(A,B,C,D))
& l1_algstr_3(g1_algstr_3(A,B,C,D)) ) ) ).
fof(free_g1_algstr_3,axiom,
! [A,B,C,D] :
( ( m1_subset_1(B,A)
& m1_subset_1(C,A)
& v1_funct_1(D)
& v1_funct_2(D,k3_zfmisc_1(A,A,A),A)
& m1_relset_1(D,k3_zfmisc_1(A,A,A),A) )
=> ! [E,F,G,H] :
( g1_algstr_3(A,B,C,D) = g1_algstr_3(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
%------------------------------------------------------------------------------