SET007 Axioms: SET007+528.ax
%------------------------------------------------------------------------------
% File : SET007+528 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Projections in n-Dimensional Euclidean Space to Each Coordinates
% 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 : jordan2b [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 48 ( 1 unt; 0 def)
% Number of atoms : 259 ( 54 equ)
% Maximal formula atoms : 19 ( 5 avg)
% Number of connectives : 226 ( 15 ~; 3 |; 62 &)
% ( 9 <=>; 137 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-3 aty)
% Number of functors : 60 ( 60 usr; 9 con; 0-4 aty)
% Number of variables : 144 ( 133 !; 11 ?)
% 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_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( D = k1_jordan2b(A,B,C)
<=> ! [E] :
( m2_finseq_1(E,k1_numbers)
=> ( E = C
=> D = k4_finseq_4(k5_numbers,k1_numbers,E,B) ) ) ) ) ) ) ) ).
fof(t1_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
& m2_relset_1(C,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
& ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),C,D) = k1_jordan2b(A,B,D) ) ) ) ) ).
fof(t2_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_finseq_1(A))
=> k1_funct_1(k5_euclid(A),B) = np__0 ) ) ) ).
fof(t3_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_finseq_1(A))
=> k1_jordan2b(A,B,k16_euclid(A)) = np__0 ) ) ) ).
fof(t4_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k1_jordan2b(A,D,k18_euclid(B,A,C)) = k3_xcmplx_0(B,k1_jordan2b(A,D,C)) ) ) ) ) ) ).
fof(t5_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(A))
=> k1_jordan2b(A,C,k19_euclid(A,B)) = k1_real_1(k1_jordan2b(A,C,B)) ) ) ) ) ).
fof(t6_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k1_jordan2b(A,D,k17_euclid(A,B,C)) = k3_real_1(k1_jordan2b(A,D,B),k1_jordan2b(A,D,C)) ) ) ) ) ) ).
fof(t7_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(A))
=> k1_jordan2b(A,D,k20_euclid(A,B,C)) = k5_real_1(k1_jordan2b(A,D,B),k1_jordan2b(A,D,C)) ) ) ) ) ) ).
fof(t8_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_finseq_1(k5_euclid(A)) = A ) ).
fof(t9_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> k16_finseq_1(k1_numbers,k5_euclid(B),A) = k5_euclid(A) ) ) ) ).
fof(t10_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_rfinseq(k1_numbers,k5_euclid(A),B) = k5_euclid(k5_binarith(A,B)) ) ) ).
fof(t11_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k15_rvsum_1(k5_euclid(A)) = np__0 ) ).
fof(t12_jordan2b,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_finseq_1(k2_funct_7(A,C,B)) = k3_finseq_1(A) ) ) ).
fof(t13_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> ! [D] :
( m1_subset_1(D,B)
=> ( r2_hidden(A,k4_finseq_1(C))
=> k3_funct_7(B,C,A,D) = k8_finseq_1(B,k8_finseq_1(B,k16_finseq_1(B,C,k5_binarith(A,np__1)),k13_binarith(B,D)),k1_rfinseq(B,C,A)) ) ) ) ) ) ).
fof(t14_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(A,k2_finseq_1(B))
=> k15_rvsum_1(k3_funct_7(k1_numbers,k5_euclid(B),A,C)) = C ) ) ) ) ).
fof(t15_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(D,k2_finseq_1(A))
& B = C )
=> ( r1_xreal_0(k1_jordan2b(A,D,C),k12_euclid(B))
& r1_xreal_0(k7_square_1(k1_jordan2b(A,D,C)),k7_square_1(k12_euclid(B))) ) ) ) ) ) ) ).
fof(t20_jordan2b,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B)))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B))) )
=> ( ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
=> ( F = k9_metric_1(B,E,D)
=> ( r1_xreal_0(D,np__0)
| v3_pre_topc(k5_pre_topc(A,k5_pcomps_1(B),C,F),A) ) ) ) ) )
=> v5_pre_topc(C,A,k5_pcomps_1(B)) ) ) ) ) ).
fof(t22_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
& m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k2_finseq_1(A))
& ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),B,D) = k1_jordan2b(A,C,D) ) )
=> v5_pre_topc(B,k15_euclid(A),k3_topmetr) ) ) ) ) ).
fof(t23_jordan2b,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k3_euclid(k13_binarith(k1_numbers,A)) = k13_binarith(k1_numbers,k18_complex1(A)) ) ).
fof(t24_jordan2b,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__1)))
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& A = k13_binarith(k1_numbers,B) ) ) ).
fof(t25_jordan2b,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k14_euclid(np__1)))
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& A = k13_binarith(k1_numbers,B) ) ) ).
fof(d2_jordan2b,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_jordan2b(A) = k9_finseq_1(A) ) ).
fof(t26_jordan2b,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k18_euclid(B,np__1,k2_jordan2b(A)) = k2_jordan2b(k3_xcmplx_0(B,A)) ) ) ).
fof(t27_jordan2b,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_jordan2b(k2_xcmplx_0(A,B)) = k17_euclid(np__1,k2_jordan2b(A),k2_jordan2b(B)) ) ) ).
fof(t28_jordan2b,axiom,
k2_jordan2b(np__0) = k16_euclid(np__1) ).
fof(t29_jordan2b,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( k2_jordan2b(A) = k2_jordan2b(B)
=> A = B ) ) ) ).
fof(t34_jordan2b,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k15_euclid(np__1)),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k15_euclid(np__1)),u1_struct_0(k3_topmetr)) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__1)))
=> k8_funct_2(u1_struct_0(k15_euclid(np__1)),u1_struct_0(k3_topmetr),A,B) = k1_jordan2b(np__1,np__1,B) )
=> v3_tops_2(A,k15_euclid(np__1),k3_topmetr) ) ) ).
fof(t35_jordan2b,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k1_jordan2b(np__2,np__1,A) = k21_euclid(A)
& k1_jordan2b(np__2,np__2,A) = k22_euclid(A) ) ) ).
fof(t36_jordan2b,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ( k1_jordan2b(np__2,np__1,A) = k8_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k16_pscomp_1,A)
& k1_jordan2b(np__2,np__2,A) = k8_funct_2(u1_struct_0(k15_euclid(np__2)),k1_numbers,k17_pscomp_1,A) ) ) ).
fof(dt_k1_jordan2b,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,u1_struct_0(k15_euclid(A))) )
=> m1_subset_1(k1_jordan2b(A,B,C),k1_numbers) ) ).
fof(dt_k2_jordan2b,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k2_jordan2b(A),u1_struct_0(k15_euclid(np__1))) ) ).
fof(t16_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( C = a_3_0_jordan2b(A,B,D)
& r2_hidden(D,k2_finseq_1(A)) )
=> v3_pre_topc(C,k15_euclid(A)) ) ) ) ) ) ).
fof(t17_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( C = a_3_1_jordan2b(A,B,D)
& r2_hidden(D,k2_finseq_1(A)) )
=> v3_pre_topc(C,k15_euclid(A)) ) ) ) ) ) ).
fof(t18_jordan2b,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] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( B = a_4_0_jordan2b(A,C,D,E)
& r2_hidden(E,k2_finseq_1(A)) )
=> v3_pre_topc(B,k15_euclid(A)) ) ) ) ) ) ) ).
fof(t19_jordan2b,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr))
& m2_relset_1(D,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) )
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),D,F) = k1_jordan2b(A,E,F) )
=> k5_pre_topc(k15_euclid(A),k3_topmetr,D,a_2_0_jordan2b(B,C)) = a_4_0_jordan2b(A,B,C,E) ) ) ) ) ) ) ).
fof(t21_jordan2b,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k8_metric_1))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( C = A
=> ( r1_xreal_0(B,np__0)
| k9_metric_1(k8_metric_1,A,B) = a_2_1_jordan2b(B,C) ) ) ) ) ) ).
fof(t30_jordan2b,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ! [B] :
( v1_xreal_0(B)
=> ( A = a_1_0_jordan2b(B)
=> v3_pre_topc(A,k3_topmetr) ) ) ) ).
fof(t31_jordan2b,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ! [B] :
( v1_xreal_0(B)
=> ( A = a_1_1_jordan2b(B)
=> v3_pre_topc(A,k3_topmetr) ) ) ) ).
fof(t32_jordan2b,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( A = a_2_0_jordan2b(B,C)
=> v3_pre_topc(A,k3_topmetr) ) ) ) ) ).
fof(t33_jordan2b,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k14_euclid(np__1)))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( k9_finseq_1(C) = A
=> ( r1_xreal_0(B,np__0)
| k9_metric_1(k14_euclid(np__1),A,B) = a_2_2_jordan2b(B,C) ) ) ) ) ) ).
fof(fraenkel_a_3_0_jordan2b,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& v1_xreal_0(C)
& m2_subset_1(D,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_3_0_jordan2b(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = E
& ~ r1_xreal_0(C,k1_jordan2b(B,D,E)) ) ) ) ).
fof(fraenkel_a_3_1_jordan2b,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& v1_xreal_0(C)
& m2_subset_1(D,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_3_1_jordan2b(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
& A = E
& ~ r1_xreal_0(k1_jordan2b(B,D,E),C) ) ) ) ).
fof(fraenkel_a_4_0_jordan2b,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& v1_xreal_0(C)
& v1_xreal_0(D)
& m2_subset_1(E,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_4_0_jordan2b(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
& A = F
& ~ r1_xreal_0(k1_jordan2b(B,E,F),C)
& ~ r1_xreal_0(D,k1_jordan2b(B,E,F)) ) ) ) ).
fof(fraenkel_a_2_0_jordan2b,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_0_jordan2b(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = D
& ~ r1_xreal_0(D,B)
& ~ r1_xreal_0(C,D) ) ) ) ).
fof(fraenkel_a_2_1_jordan2b,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_1_jordan2b(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = D
& ~ r1_xreal_0(D,k6_xcmplx_0(C,B))
& ~ r1_xreal_0(k2_xcmplx_0(C,B),D) ) ) ) ).
fof(fraenkel_a_1_0_jordan2b,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r2_hidden(A,a_1_0_jordan2b(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& ~ r1_xreal_0(B,C) ) ) ) ).
fof(fraenkel_a_1_1_jordan2b,axiom,
! [A,B] :
( v1_xreal_0(B)
=> ( r2_hidden(A,a_1_1_jordan2b(B))
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& A = C
& ~ r1_xreal_0(C,B) ) ) ) ).
fof(fraenkel_a_2_2_jordan2b,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_2_jordan2b(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = k13_binarith(k1_numbers,D)
& ~ r1_xreal_0(D,k6_xcmplx_0(C,B))
& ~ r1_xreal_0(k2_xcmplx_0(C,B),D) ) ) ) ).
%------------------------------------------------------------------------------