SET007 Axioms: SET007+259.ax
%------------------------------------------------------------------------------
% File : SET007+259 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Homotheties and Shears in Affine Planes
% 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 : homothet [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 10 ( 0 unt; 0 def)
% Number of atoms : 202 ( 24 equ)
% Maximal formula atoms : 31 ( 20 avg)
% Number of connectives : 233 ( 41 ~; 8 |; 114 &)
% ( 3 <=>; 67 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 20 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-5 aty)
% Number of functors : 4 ( 4 usr; 0 con; 1-4 aty)
% Number of variables : 60 ( 60 !; 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(t1_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(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))
=> ( ( r1_aff_1(A,B,C,E)
& r1_aff_1(A,B,C,F)
& r1_aff_1(A,B,C,G)
& r1_aff_1(A,B,D,H)
& r1_aff_1(A,B,D,I)
& r1_aff_1(A,B,D,J)
& r2_analoaf(A,C,D,E,H)
& r2_analoaf(A,C,I,E,J)
& r2_analoaf(A,F,D,G,H)
& v4_aff_2(A) )
=> ( r1_aff_1(A,B,C,D)
| D = I
| C = F
| B = I
| B = F
| r2_analoaf(A,F,I,G,J) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t2_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(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))
=> ~ ( B != C
& B != D
& r1_aff_1(A,B,C,D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( v6_transgeo(E,A)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B) = B
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,C) = D ) ) ) ) ) )
=> v4_aff_2(A) ) ) ).
fof(t3_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(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))
=> ~ ( B != C
& B != D
& r1_aff_1(A,B,C,D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( v6_transgeo(E,A)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B) = B
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,C) = D ) ) ) ) ) ) ) ) ).
fof(t4_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v4_aff_2(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))
=> ~ ( B != C
& B != D
& r1_aff_1(A,B,C,D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( v6_transgeo(E,A)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B) = B
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,C) = D ) ) ) ) ) ) ) ) ).
fof(d1_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_homothet(A,B,C)
<=> ( v8_transgeo(B,A)
& v1_aff_1(C,A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D) = D ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r2_aff_1(A,D,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D),C) ) ) ) ) ) ) ).
fof(t5_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> ( ( r1_homothet(A,D,C)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,B) = B )
=> ( r2_hidden(B,C)
| D = k6_partfun1(u1_struct_0(A)) ) ) ) ) ) ) ).
fof(t6_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_aff_1(A,B,C,D)
& ~ r2_hidden(B,D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( r1_homothet(A,E,D)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B) = C ) ) ) ) ) )
=> v7_aff_2(A) ) ) ).
fof(t7_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(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(C,J)
& r2_hidden(D,J)
& r2_hidden(E,J)
& v7_aff_2(A)
& r2_hidden(F,K)
& r2_hidden(G,K)
& r2_hidden(H,K)
& r2_hidden(I,K)
& r2_analoaf(A,B,F,D,H)
& r2_analoaf(A,B,G,D,I)
& r2_analoaf(A,C,F,E,H) )
=> ( F = G
| C = B
| r2_analoaf(A,C,G,E,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_aff_1(A,B,C,D)
& ~ r2_hidden(B,D)
& v7_aff_2(A)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( r1_homothet(A,E,D)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B) = C ) ) ) ) ) ) ) ).
fof(t9_homothet,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_realset2(A)
& v1_diraf(A)
& v2_diraf(A)
& l1_analoaf(A) )
=> ( v7_aff_2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_aff_1(A,B,C,D)
& ~ r2_hidden(B,D)
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& v3_funct_2(E,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
=> ~ ( r1_homothet(A,E,D)
& k8_funct_2(u1_struct_0(A),u1_struct_0(A),E,B) = C ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------