SET007 Axioms: SET007+787.ax
%------------------------------------------------------------------------------
% File : SET007+787 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Cross Products and Tripple Vector Products in 3-dimensiona
% 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 : euclid_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 47 ( 1 unt; 0 def)
% Number of atoms : 196 ( 58 equ)
% Maximal formula atoms : 8 ( 4 avg)
% Number of connectives : 149 ( 0 ~; 0 |; 23 &)
% ( 3 <=>; 123 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 7 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 29 ( 29 usr; 6 con; 0-4 aty)
% Number of variables : 126 ( 123 !; 3 ?)
% 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_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ? [B] :
( m1_subset_1(B,k1_numbers)
& ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& A = k3_finseq_4(k1_numbers,B,C,D) ) ) ) ) ).
fof(d1_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k1_euclid_5(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( A = C
=> B = k1_funct_1(C,np__1) ) ) ) ) ) ).
fof(d2_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k2_euclid_5(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( A = C
=> B = k1_funct_1(C,np__2) ) ) ) ) ) ).
fof(d3_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B = k3_euclid_5(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( A = C
=> B = k1_funct_1(C,np__3) ) ) ) ) ) ).
fof(d4_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k4_euclid_5(A,B,C) = k3_finseq_4(k1_numbers,A,B,C) ) ) ) ).
fof(t2_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( k1_euclid_5(k4_euclid_5(A,B,C)) = A
& k2_euclid_5(k4_euclid_5(A,B,C)) = B
& k3_euclid_5(k4_euclid_5(A,B,C)) = C ) ) ) ) ).
fof(t3_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> A = k4_euclid_5(k1_euclid_5(A),k2_euclid_5(A),k3_euclid_5(A)) ) ).
fof(t4_euclid_5,axiom,
k16_euclid(np__3) = k4_euclid_5(np__0,np__0,np__0) ).
fof(t5_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k17_euclid(np__3,A,B) = k4_euclid_5(k9_binop_2(k1_euclid_5(A),k1_euclid_5(B)),k9_binop_2(k2_euclid_5(A),k2_euclid_5(B)),k9_binop_2(k3_euclid_5(A),k3_euclid_5(B))) ) ) ).
fof(t6_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> k17_euclid(np__3,k4_euclid_5(A,B,C),k4_euclid_5(D,E,F)) = k4_euclid_5(k9_binop_2(A,D),k9_binop_2(B,E),k9_binop_2(C,F)) ) ) ) ) ) ) ).
fof(t7_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k18_euclid(A,np__3,B) = k4_euclid_5(k11_binop_2(A,k1_euclid_5(B)),k11_binop_2(A,k2_euclid_5(B)),k11_binop_2(A,k3_euclid_5(B))) ) ) ).
fof(t8_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k18_euclid(A,np__3,k4_euclid_5(B,C,D)) = k4_euclid_5(k11_binop_2(A,B),k11_binop_2(A,C),k11_binop_2(A,D)) ) ) ) ) ).
fof(t9_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ( k1_euclid_5(k18_euclid(A,np__3,B)) = k11_binop_2(A,k1_euclid_5(B))
& k2_euclid_5(k18_euclid(A,np__3,B)) = k11_binop_2(A,k2_euclid_5(B))
& k3_euclid_5(k18_euclid(A,np__3,B)) = k11_binop_2(A,k3_euclid_5(B)) ) ) ) ).
fof(t10_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> k19_euclid(np__3,A) = k4_euclid_5(k7_binop_2(k1_euclid_5(A)),k7_binop_2(k2_euclid_5(A)),k7_binop_2(k3_euclid_5(A))) ) ).
fof(t11_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k19_euclid(np__3,k4_euclid_5(A,B,C)) = k4_euclid_5(k7_binop_2(A),k7_binop_2(B),k7_binop_2(C)) ) ) ) ).
fof(t12_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k20_euclid(np__3,A,B) = k4_euclid_5(k10_binop_2(k1_euclid_5(A),k1_euclid_5(B)),k10_binop_2(k2_euclid_5(A),k2_euclid_5(B)),k10_binop_2(k3_euclid_5(A),k3_euclid_5(B))) ) ) ).
fof(t13_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> k20_euclid(np__3,k4_euclid_5(A,B,C),k4_euclid_5(D,E,F)) = k4_euclid_5(k10_binop_2(A,D),k10_binop_2(B,E),k10_binop_2(C,F)) ) ) ) ) ) ) ).
fof(d5_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(A,B) = k4_euclid_5(k10_binop_2(k11_binop_2(k2_euclid_5(A),k3_euclid_5(B)),k11_binop_2(k3_euclid_5(A),k2_euclid_5(B))),k10_binop_2(k11_binop_2(k3_euclid_5(A),k1_euclid_5(B)),k11_binop_2(k1_euclid_5(A),k3_euclid_5(B))),k10_binop_2(k11_binop_2(k1_euclid_5(A),k2_euclid_5(B)),k11_binop_2(k2_euclid_5(A),k1_euclid_5(B)))) ) ) ).
fof(t14_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__3)))
=> ( D = k4_euclid_5(A,B,C)
=> ( k1_euclid_5(D) = A
& k2_euclid_5(D) = B
& k3_euclid_5(D) = C ) ) ) ) ) ) ).
fof(t15_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> k5_euclid_5(k4_euclid_5(A,B,C),k4_euclid_5(D,E,F)) = k4_euclid_5(k10_binop_2(k11_binop_2(B,F),k11_binop_2(C,E)),k10_binop_2(k11_binop_2(C,D),k11_binop_2(A,F)),k10_binop_2(k11_binop_2(A,E),k11_binop_2(B,D))) ) ) ) ) ) ) ).
fof(t16_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> ( k5_euclid_5(k18_euclid(A,np__3,B),C) = k18_euclid(A,np__3,k5_euclid_5(B,C))
& k5_euclid_5(k18_euclid(A,np__3,B),C) = k5_euclid_5(B,k18_euclid(A,np__3,C)) ) ) ) ) ).
fof(t17_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(A,B) = k19_euclid(np__3,k5_euclid_5(B,A)) ) ) ).
fof(t18_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(k19_euclid(np__3,A),B) = k5_euclid_5(A,k19_euclid(np__3,B)) ) ) ).
fof(t19_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> k5_euclid_5(k4_euclid_5(np__0,np__0,np__0),k4_euclid_5(A,B,C)) = k16_euclid(np__3) ) ) ) ).
fof(t20_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_euclid_5(k4_euclid_5(A,np__0,np__0),k4_euclid_5(B,np__0,np__0)) = k16_euclid(np__3) ) ) ).
fof(t21_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_euclid_5(k4_euclid_5(np__0,A,np__0),k4_euclid_5(np__0,B,np__0)) = k16_euclid(np__3) ) ) ).
fof(t22_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_euclid_5(k4_euclid_5(np__0,np__0,A),k4_euclid_5(np__0,np__0,B)) = k16_euclid(np__3) ) ) ).
fof(t23_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(A,k17_euclid(np__3,B,C)) = k17_euclid(np__3,k5_euclid_5(A,B),k5_euclid_5(A,C)) ) ) ) ).
fof(t24_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(k17_euclid(np__3,A,B),C) = k17_euclid(np__3,k5_euclid_5(A,C),k5_euclid_5(B,C)) ) ) ) ).
fof(t25_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(A,A) = k16_euclid(np__3) ) ).
fof(t26_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(k17_euclid(np__3,A,B),k17_euclid(np__3,C,D)) = k17_euclid(np__3,k17_euclid(np__3,k17_euclid(np__3,k5_euclid_5(A,C),k5_euclid_5(A,D)),k5_euclid_5(B,C)),k5_euclid_5(B,D)) ) ) ) ) ).
fof(t27_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> A = k3_finseq_4(k1_numbers,k1_euclid_5(A),k2_euclid_5(A),k3_euclid_5(A)) ) ).
fof(t28_euclid_5,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( k3_finseq_1(A) = np__3
& k3_finseq_1(B) = np__3 )
=> k13_rvsum_1(A,B) = k3_finseq_4(k1_numbers,k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,A,np__1),k2_seq_1(k5_numbers,k1_numbers,B,np__1)),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,A,np__2),k2_seq_1(k5_numbers,k1_numbers,B,np__2)),k11_binop_2(k2_seq_1(k5_numbers,k1_numbers,A,np__3),k2_seq_1(k5_numbers,k1_numbers,B,np__3))) ) ) ) ).
fof(t29_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> k2_euclid_2(np__3,A,B) = k9_binop_2(k9_binop_2(k11_binop_2(k1_euclid_5(A),k1_euclid_5(B)),k11_binop_2(k2_euclid_5(A),k2_euclid_5(B))),k11_binop_2(k3_euclid_5(A),k3_euclid_5(B))) ) ) ).
fof(t30_euclid_5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> k2_euclid_2(np__3,k4_euclid_5(C,D,A),k4_euclid_5(E,F,B)) = k9_binop_2(k9_binop_2(k11_binop_2(C,E),k11_binop_2(D,F)),k11_binop_2(A,B)) ) ) ) ) ) ) ).
fof(d6_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k6_euclid_5(A,B,C) = k2_euclid_2(np__3,A,k5_euclid_5(B,C)) ) ) ) ).
fof(t31_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ( k6_euclid_5(A,A,B) = np__0
& k6_euclid_5(B,A,B) = np__0 ) ) ) ).
fof(t32_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k5_euclid_5(A,k5_euclid_5(B,C)) = k20_euclid(np__3,k18_euclid(k2_euclid_2(np__3,A,C),np__3,B),k18_euclid(k2_euclid_2(np__3,A,B),np__3,C)) ) ) ) ).
fof(t33_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k6_euclid_5(A,B,C) = k6_euclid_5(B,C,A) ) ) ) ).
fof(t34_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k6_euclid_5(A,B,C) = k6_euclid_5(C,A,B) ) ) ) ).
fof(t35_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__3)))
=> k6_euclid_5(A,B,C) = k2_euclid_2(np__3,k5_euclid_5(A,B),C) ) ) ) ).
fof(dt_k1_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> m1_subset_1(k1_euclid_5(A),k1_numbers) ) ).
fof(dt_k2_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> m1_subset_1(k2_euclid_5(A),k1_numbers) ) ).
fof(dt_k3_euclid_5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
=> m1_subset_1(k3_euclid_5(A),k1_numbers) ) ).
fof(dt_k4_euclid_5,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_numbers) )
=> m1_subset_1(k4_euclid_5(A,B,C),u1_struct_0(k15_euclid(np__3))) ) ).
fof(dt_k5_euclid_5,axiom,
! [A,B] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__3))) )
=> m1_subset_1(k5_euclid_5(A,B),u1_struct_0(k15_euclid(np__3))) ) ).
fof(dt_k6_euclid_5,axiom,
! [A,B,C] :
( ( m1_subset_1(A,u1_struct_0(k15_euclid(np__3)))
& m1_subset_1(B,u1_struct_0(k15_euclid(np__3)))
& m1_subset_1(C,u1_struct_0(k15_euclid(np__3))) )
=> v1_xreal_0(k6_euclid_5(A,B,C)) ) ).
%------------------------------------------------------------------------------