%------------------------------------------------------------------------------
% File : SET007+296 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Shear Theorems and Their Role in Affine Geometry
% 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 : conmetr1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 23 ( 0 unt; 0 def)
% Number of atoms : 343 ( 0 equ)
% Maximal formula atoms : 41 ( 14 avg)
% Number of connectives : 361 ( 41 ~; 48 |; 158 &)
% ( 17 <=>; 97 =>; 0 <=; 0 <~>)
% Maximal formula depth : 41 ( 15 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 29 ( 29 usr; 0 prp; 1-5 aty)
% Number of functors : 3 ( 3 usr; 0 con; 1-1 aty)
% Number of variables : 85 ( 85 !; 0 ?)
% 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(d1_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v1_conmetr1(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))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r4_aff_1(A,J,K)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(F,J)
& r2_hidden(H,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(G,K)
& r2_hidden(I,K)
& r2_analoaf(A,D,C,H,G)
& r2_analoaf(A,C,B,G,F)
& r2_analoaf(A,B,E,F,I) )
=> ( r2_hidden(E,J)
| r2_hidden(C,J)
| r2_hidden(G,J)
| r2_hidden(I,J)
| r2_hidden(B,K)
| r2_hidden(D,K)
| r2_hidden(F,K)
| r2_hidden(H,K)
| r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d2_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v2_conmetr1(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))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ! [L] :
( m1_subset_1(L,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(K,A)
& v1_aff_1(L,A)
& r2_hidden(B,K)
& r2_hidden(B,L)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(G,K)
& r2_hidden(I,K)
& r2_hidden(D,L)
& r2_hidden(F,L)
& r2_hidden(H,L)
& r2_hidden(J,L)
& r2_analoaf(A,E,D,I,H)
& r2_analoaf(A,D,C,H,G)
& r2_analoaf(A,C,F,G,J) )
=> ( r2_hidden(F,K)
| r2_hidden(D,K)
| r2_hidden(H,K)
| r2_hidden(J,K)
| r2_hidden(C,L)
| r2_hidden(E,L)
| r2_hidden(G,L)
| r2_hidden(I,L)
| r2_analoaf(A,E,F,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d3_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v3_conmetr1(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))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(J,A)
& v1_aff_1(K,A)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(F,J)
& r2_hidden(H,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(G,K)
& r2_hidden(I,K)
& r2_analoaf(A,D,C,H,G)
& r2_analoaf(A,C,B,G,F)
& r2_analoaf(A,B,E,F,I) )
=> ( r2_hidden(E,J)
| r2_hidden(C,J)
| r2_hidden(G,J)
| r2_hidden(I,J)
| r2_hidden(B,K)
| r2_hidden(D,K)
| r2_hidden(F,K)
| r2_hidden(H,K)
| r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_conmetr1(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))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(J,A)
& v1_aff_1(K,A)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(G,J)
& r2_hidden(I,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(F,K)
& r2_hidden(H,K)
& r2_analoaf(A,D,C,H,G)
& r2_analoaf(A,C,B,G,F)
& r2_analoaf(A,B,E,F,I) )
=> ( r2_hidden(E,J)
| r2_hidden(C,J)
| r2_hidden(F,J)
| r2_hidden(H,J)
| r2_hidden(B,K)
| r2_hidden(D,K)
| r2_hidden(G,K)
| r2_hidden(I,K)
| r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v5_conmetr1(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))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r4_aff_1(A,J,K)
& r2_hidden(B,J)
& r2_hidden(D,J)
& r2_hidden(G,J)
& r2_hidden(I,J)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(F,K)
& r2_hidden(H,K)
& r2_analoaf(A,D,C,H,G)
& r2_analoaf(A,C,B,G,F)
& r2_analoaf(A,B,E,F,I) )
=> ( r2_hidden(E,J)
| r2_hidden(C,J)
| r2_hidden(F,J)
| r2_hidden(H,J)
| r2_hidden(B,K)
| r2_hidden(D,K)
| r2_hidden(G,K)
| r2_hidden(I,K)
| r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v6_conmetr1(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))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(A))
=> ! [K] :
( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
=> ! [L] :
( m1_subset_1(L,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( v1_aff_1(K,A)
& v1_aff_1(L,A)
& r2_hidden(B,K)
& r2_hidden(B,L)
& r2_hidden(C,K)
& r2_hidden(E,K)
& r2_hidden(H,K)
& r2_hidden(J,K)
& r2_hidden(D,L)
& r2_hidden(F,L)
& r2_hidden(G,L)
& r2_hidden(I,L)
& r2_analoaf(A,E,D,I,H)
& r2_analoaf(A,D,C,H,G)
& r2_analoaf(A,C,F,G,J) )
=> ( r2_hidden(F,K)
| r2_hidden(D,K)
| r2_hidden(G,K)
| r2_hidden(I,K)
| r2_hidden(C,L)
| r2_hidden(E,L)
| r2_hidden(H,L)
| r2_hidden(J,L)
| r2_analoaf(A,E,F,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t1_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_conmetr1(A)
<=> ( v5_conmetr1(A)
& v6_conmetr1(A) ) ) ) ).
fof(t2_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v3_conmetr1(A)
<=> ( v1_conmetr1(A)
& v2_conmetr1(A) ) ) ) ).
fof(t3_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v5_conmetr1(A)
=> v1_conmetr1(A) ) ) ).
fof(t4_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v6_conmetr1(A)
=> v2_conmetr1(A) ) ) ).
fof(t5_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_conmetr1(A)
=> v3_conmetr1(A) ) ) ).
fof(t6_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v11_aff_2(A)
=> v1_conmetr1(A) ) ) ).
fof(t7_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(A)
=> v2_conmetr1(A) ) ) ).
fof(t8_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(A)
<=> v3_conmetr1(A) ) ) ).
fof(t9_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v13_aff_2(A)
<=> v5_conmetr1(A) ) ) ).
fof(t10_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v2_aff_2(A)
<=> v6_conmetr1(A) ) ) ).
fof(t11_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v1_aff_2(A)
<=> v4_conmetr1(A) ) ) ).
fof(t12_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v6_conmetr1(A)
=> v5_conmetr1(A) ) ) ).
fof(t13_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v3_conmetr1(k3_analmetr(A))
<=> v6_conmetr(A) ) ) ).
fof(t14_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v5_conmetr(A)
<=> v7_aff_2(k3_analmetr(A)) ) ) ).
fof(t15_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v11_aff_2(k3_analmetr(A))
<=> v8_conmetr(A) ) ) ).
fof(t16_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v2_conmetr(A)
<=> v2_aff_2(k3_analmetr(A)) ) ) ).
fof(t17_conmetr1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_analmetr(A)
& l1_analmetr(A) )
=> ( v1_conaffm(A)
<=> v4_aff_2(k3_analmetr(A)) ) ) ).
%------------------------------------------------------------------------------