SET007 Axioms: SET007+919.ax


%------------------------------------------------------------------------------
% File     : SET007+919 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Homeomorphisms of Jordan Curves
% 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    : jordan24 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   24 (   0 unit)
%            Number of atoms       :  227 (  10 equality)
%            Maximal formula depth :   18 (  10 average)
%            Number of connectives :  222 (  19 ~  ;   2  |; 104  &)
%                                         (   6 <=>;  91 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   39 (   0 propositional; 1-4 arity)
%            Number of functors    :   32 (   8 constant; 0-4 arity)
%            Number of variables   :   77 (   0 singleton;  72 !;   5 ?)
%            Maximal term depth    :    7 (   2 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(rc1_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ? [B] :
          ( m1_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
          & v1_jordan24(B,A) ) ) )).

fof(cc1_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
         => ( ( v1_funct_1(B)
              & v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v1_jordan24(B,A) )
           => ( v1_funct_1(B)
              & v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v5_pre_topc(B,k5_pcomps_1(A),k5_pcomps_1(A)) ) ) ) ) )).

fof(cc2_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
         => ( ( v1_funct_1(B)
              & v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v1_jordan24(B,A) )
           => ( v1_funct_1(B)
              & v2_funct_1(B)
              & ~ v1_xboole_0(B)
              & v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v2_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v3_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v5_pre_topc(B,k5_pcomps_1(A),k5_pcomps_1(A))
              & v1_partfun1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
              & v1_t_0topsp(B,k5_pcomps_1(A),k5_pcomps_1(A))
              & v3_tops_2(B,k5_pcomps_1(A),k5_pcomps_1(A)) ) ) ) ) )).

fof(d1_jordan24,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ( r1_jordan24(A,B,C,D)
                  <=> ( r2_hidden(C,B)
                      & r2_hidden(D,B)
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
                             => ( ( r2_hidden(E,B)
                                  & r2_hidden(F,B) )
                               => r1_xreal_0(k1_gobrd14(A,E,F),k1_gobrd14(A,C,D)) ) ) ) ) ) ) ) ) ) )).

fof(t1_jordan24,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
          & ? [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
              & r1_jordan24(np__2,A,B,C) ) ) ) )).

fof(d2_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A)))
            & m2_relset_1(B,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(A))) )
         => ( v1_jordan24(B,A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & v3_vectmetr(C,A)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A))
                & C = B ) ) ) ) )).

fof(d3_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)))
            & m2_relset_1(B,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2))) )
         => ( B = k1_jordan24(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
               => k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),B,C) = k23_euclid(k3_complex1(k3_complex2(k2_xcmplx_0(k21_euclid(C),k3_xcmplx_0(k22_euclid(C),k1_xcmplx_0)),A)),k4_complex1(k3_complex2(k2_xcmplx_0(k21_euclid(C),k3_xcmplx_0(k22_euclid(C),k1_xcmplx_0)),A))) ) ) ) ) )).

fof(t2_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( r1_xreal_0(np__0,A)
       => ( r1_xreal_0(k4_real_1(np__2,k32_sin_cos),A)
          | ! [B] :
              ( ( v1_funct_1(B)
                & v1_funct_2(B,u1_struct_0(k5_pcomps_1(k14_euclid(np__2))),u1_struct_0(k5_pcomps_1(k14_euclid(np__2))))
                & m2_relset_1(B,u1_struct_0(k5_pcomps_1(k14_euclid(np__2))),u1_struct_0(k5_pcomps_1(k14_euclid(np__2)))) )
             => ( B = k1_jordan24(A)
               => v1_jordan24(B,k14_euclid(np__2)) ) ) ) ) ) )).

fof(t3_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( v1_xreal_0(E)
                     => ! [F] :
                          ( v1_xreal_0(F)
                         => ( r1_jordan24(np__2,C,A,B)
                           => r1_jordan24(np__2,k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),k2_jgraph_2(D,E,D,F),C),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(D,E,D,F),A),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k2_jgraph_2(D,E,D,F),B)) ) ) ) ) ) ) ) )).

fof(t4_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r1_xreal_0(np__0,D)
                      & r1_jordan24(np__2,C,A,B) )
                   => ( r1_xreal_0(k4_real_1(np__2,k32_sin_cos),D)
                      | r1_jordan24(np__2,k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),k1_jordan24(D),C),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k1_jordan24(D),A),k8_funct_2(u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)),k1_jordan24(D),B)) ) ) ) ) ) ) )).

fof(t5_jordan24,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k3_complex2(A,k1_real_1(B)) = k3_complex2(A,k5_real_1(k4_real_1(np__2,k32_sin_cos),B)) ) ) )).

fof(t6_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k1_jordan24(k1_real_1(A)) = k1_jordan24(k5_real_1(k4_real_1(np__2,k32_sin_cos),A)) ) )).

fof(t7_jordan24,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ? [B] :
          ( m1_topgrp_1(B,k15_euclid(np__2))
          & r1_jordan24(np__2,k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),B,A),k23_euclid(k1_real_1(np__1),np__0),k23_euclid(np__1,np__0)) ) ) )).

fof(d4_jordan24,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( l1_pre_topc(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v2_jordan24(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( v4_pre_topc(D,A)
                     => v4_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) )).

fof(t8_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( ( v2_funct_1(C)
                  & v2_funct_2(C,u1_struct_0(A),u1_struct_0(B)) )
               => ( v3_tops_2(C,A,B)
                <=> v2_jordan24(C,A,B) ) ) ) ) ) )).

fof(t9_jordan24,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ( k3_subset_1(A,B) = k1_xboole_0
      <=> B = A ) ) )).

fof(t10_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( v2_connsp_1(D,A)
                     => v2_connsp_1(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) )).

fof(t11_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r3_connsp_1(A,D)
                     => r3_connsp_1(B,k4_pre_topc(A,B,C,D)) ) ) ) ) ) ) )).

fof(t12_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( v1_funct_1(k7_relat_1(C,D))
                    & v1_funct_2(k7_relat_1(C,D),u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D))))
                    & m2_relset_1(k7_relat_1(C,D),u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D)))) ) ) ) ) ) )).

fof(t13_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D))))
                          & m2_relset_1(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D)))) )
                       => ( E = k7_relat_1(C,D)
                         => v5_pre_topc(E,k3_pre_topc(A,D),k3_pre_topc(B,k4_pre_topc(A,B,C,D))) ) ) ) ) ) ) ) )).

fof(t14_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D))))
                          & m2_relset_1(E,u1_struct_0(k3_pre_topc(A,D)),u1_struct_0(k3_pre_topc(B,k4_pre_topc(A,B,C,D)))) )
                       => ( E = k7_relat_1(C,D)
                         => v3_tops_2(E,k3_pre_topc(A,D),k3_pre_topc(B,k4_pre_topc(A,B,C,D))) ) ) ) ) ) ) ) )).

fof(t15_jordan24,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v3_tops_2(C,A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                       => ( r4_connsp_1(A,E,D)
                         => r4_connsp_1(B,k4_pre_topc(A,B,C,E),k4_pre_topc(A,B,C,D)) ) ) ) ) ) ) ) )).

fof(t16_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m1_topgrp_1(B,k15_euclid(np__2))
         => ( v2_jordan1(A)
           => v2_jordan1(k4_pre_topc(k15_euclid(np__2),k15_euclid(np__2),B,A)) ) ) ) )).

fof(dt_k1_jordan24,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( v1_funct_1(k1_jordan24(A))
        & v1_funct_2(k1_jordan24(A),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2)))
        & m2_relset_1(k1_jordan24(A),u1_struct_0(k15_euclid(np__2)),u1_struct_0(k15_euclid(np__2))) ) ) )).
%------------------------------------------------------------------------------