SET007 Axioms: SET007+259.ax


%------------------------------------------------------------------------------
% File     : SET007+259 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  202 (  24 equality)
%            Maximal formula depth :   36 (  20 average)
%            Number of connectives :  233 (  41 ~  ;   8  |; 114  &)
%                                         (   3 <=>;  67 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-5 arity)
%            Number of functors    :    4 (   0 constant; 1-4 arity)
%            Number of variables   :   60 (   0 singleton;  60 !;   0 ?)
%            Maximal term depth    :    3 (   1 average)
% 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 ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------