SET007 Axioms: SET007+921.ax
%------------------------------------------------------------------------------
% File : SET007+921 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Inner Product and Conjugate of Matrix of Complex Numbers
% 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 : matrixc1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 0 unt; 0 def)
% Number of atoms : 501 ( 161 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 410 ( 4 ~; 15 |; 129 &)
% ( 7 <=>; 255 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 66 ( 66 usr; 9 con; 0-6 aty)
% Number of variables : 213 ( 212 !; 1 ?)
% 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_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( B = k1_matrixc1(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& k1_matrix_1(B) = k1_matrix_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(C,D),k2_matrix_1(A))
=> k3_matrix_1(k2_numbers,B,C,D) = k15_complex1(k3_matrix_1(k2_numbers,A,C,D)) ) ) ) ) ) ) ) ).
fof(t1_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( r2_hidden(k4_tarski(A,B),k2_matrix_1(C))
<=> ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,k3_finseq_1(C))
& r1_xreal_0(np__1,B)
& r1_xreal_0(B,k1_matrix_1(C)) ) ) ) ) ) ).
fof(t2_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k1_matrixc1(k1_matrixc1(A)) = A ) ).
fof(t3_matrixc1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( k3_finseq_1(k7_matrix_5(A,B)) = k3_finseq_1(B)
& k1_matrix_1(k7_matrix_5(A,B)) = k1_matrix_1(B) ) ) ) ).
fof(t4_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( v1_xcmplx_0(C)
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(k7_matrix_5(C,D)) = k3_finseq_1(D)
& k1_matrix_1(k7_matrix_5(C,D)) = k1_matrix_1(D)
& r2_hidden(k4_tarski(A,B),k2_matrix_1(D)) )
=> k3_matrix_1(k2_numbers,k7_matrix_5(C,D),A,B) = k3_xcmplx_0(C,k3_matrix_1(k2_numbers,D,A,B)) ) ) ) ) ) ).
fof(t5_matrixc1,axiom,
! [A] :
( v1_xcmplx_0(A)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> k1_matrixc1(k7_matrix_5(A,B)) = k7_matrix_5(k15_complex1(A),k1_matrixc1(B)) ) ) ).
fof(t6_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( k3_finseq_1(k3_matrix_5(A,B)) = k3_finseq_1(A)
& k1_matrix_1(k3_matrix_5(A,B)) = k1_matrix_1(A) ) ) ) ).
fof(t7_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k1_matrix_1(C) = k1_matrix_1(D)
& r2_hidden(k4_tarski(A,B),k2_matrix_1(C)) )
=> k3_matrix_1(k2_numbers,k3_matrix_5(C,D),A,B) = k3_binop_2(k3_matrix_1(k2_numbers,C,A,B),k3_matrix_1(k2_numbers,D,A,B)) ) ) ) ) ) ).
fof(t8_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> k1_matrixc1(k3_matrix_5(A,B)) = k3_matrix_5(k1_matrixc1(A),k1_matrixc1(B)) ) ) ) ).
fof(t9_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( k3_finseq_1(k4_matrix_5(A)) = k3_finseq_1(A)
& k1_matrix_1(k4_matrix_5(A)) = k1_matrix_1(A) ) ) ).
fof(t10_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(k4_matrix_5(C)) = k3_finseq_1(C)
& k1_matrix_1(k4_matrix_5(C)) = k1_matrix_1(C)
& r2_hidden(k4_tarski(A,B),k2_matrix_1(C)) )
=> k3_matrix_1(k2_numbers,k4_matrix_5(C),A,B) = k1_binop_2(k3_matrix_1(k2_numbers,C,A,B)) ) ) ) ) ).
fof(t11_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k7_matrix_5(k7_binop_2(np__1),A) = k4_matrix_5(A) ) ).
fof(t12_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k1_matrixc1(k4_matrix_5(A)) = k4_matrix_5(k1_matrixc1(A)) ) ).
fof(t13_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( k3_finseq_1(k5_matrix_5(A,B)) = k3_finseq_1(A)
& k1_matrix_1(k5_matrix_5(A,B)) = k1_matrix_1(A) ) ) ) ).
fof(t14_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k1_matrix_1(C) = k1_matrix_1(D)
& r2_hidden(k4_tarski(A,B),k2_matrix_1(C)) )
=> k3_matrix_1(k2_numbers,k5_matrix_5(C,D),A,B) = k4_binop_2(k3_matrix_1(k2_numbers,C,A,B),k3_matrix_1(k2_numbers,D,A,B)) ) ) ) ) ) ).
fof(t15_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> k1_matrixc1(k5_matrix_5(A,B)) = k5_matrix_5(k1_matrixc1(A),k1_matrixc1(B)) ) ) ) ).
fof(d2_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k2_matrixc1(A) = k1_matrixc1(k4_matrix_1(k2_numbers,A)) ) ).
fof(d3_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( B = k3_matrixc1(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& k1_matrix_1(B) = np__1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(k3_finseq_1(A)))
=> k1_funct_1(B,C) = k13_binarith(k2_numbers,k9_matrix_5(A,C)) ) ) ) ) ) ) ) ).
fof(d4_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k4_matrixc1(A) = k8_matrix_1(k2_numbers,A,np__1) ) ).
fof(d5_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> k5_matrixc1(A,B) = k1_finseqop(k2_numbers,k2_numbers,k2_numbers,k29_binop_2,A,B) ) ) ).
fof(d6_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k6_matrixc1(A) = k2_finsop_1(k2_numbers,A,k27_binop_2) ) ).
fof(d7_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( C = k7_matrixc1(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_finseq_1(k3_finseq_1(A)))
=> k9_matrix_5(C,D) = k6_matrixc1(k5_matrixc1(k7_matrix_1(k2_numbers,A,D),B)) ) ) ) ) ) ) ) ).
fof(t16_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
=> ! [D] :
( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
=> k6_complsp1(B,k5_matrixc1(C,D)) = k5_matrixc1(k4_complsp2(A,B,C),D) ) ) ) ) ).
fof(d8_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( v1_xcmplx_0(B)
=> k8_matrixc1(A,B) = k7_matrix_5(B,A) ) ) ).
fof(t17_matrixc1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> k1_matrixc1(k8_matrixc1(B,A)) = k7_matrix_5(k15_complex1(A),k1_matrixc1(B)) ) ) ).
fof(t18_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( k3_finseq_1(k5_matrixc1(A,B)) = k3_finseq_1(A)
& k3_finseq_1(k5_matrixc1(A,B)) = k3_finseq_1(B) ) ) ) ) ).
fof(t19_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(k5_matrixc1(A,B)))
=> k9_matrix_5(k5_matrixc1(A,B),C) = k5_binop_2(k9_matrix_5(A,C),k9_matrix_5(B,C)) ) ) ) ) ).
fof(t20_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
=> ! [D] :
( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
=> k9_matrix_5(k9_matrixc1(A,C,D),B) = k5_binop_2(k9_matrix_5(C,B),k9_matrix_5(D,B)) ) ) ) ) ).
fof(t21_matrixc1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k15_complex1(k2_binop_1(k2_numbers,k2_numbers,k2_numbers,k27_binop_2,A,k15_complex1(B))) = k2_binop_1(k2_numbers,k2_numbers,k2_numbers,k27_binop_2,k15_complex1(A),B) ) ) ).
fof(t22_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_numbers)
& m2_relset_1(B,k5_numbers,k2_numbers)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(A)) )
=> k8_funct_2(k5_numbers,k2_numbers,B,C) = k9_matrix_5(A,C) ) ) ) ) ).
fof(t23_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ( r1_xreal_0(np__1,k3_finseq_1(k1_complsp2(A)))
=> k2_finsop_1(k2_numbers,k1_complsp2(A),k27_binop_2) = k15_complex1(k2_finsop_1(k2_numbers,A,k27_binop_2)) ) ) ).
fof(t24_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ( r1_xreal_0(np__1,k3_finseq_1(A))
=> k6_matrixc1(k1_complsp2(A)) = k15_complex1(k6_matrixc1(A)) ) ) ).
fof(t25_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k1_complsp2(k5_matrixc1(A,k1_complsp2(B))) = k5_matrixc1(B,k1_complsp2(A)) ) ) ) ).
fof(t26_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k5_matrixc1(A,k6_complsp1(C,B)) = k6_complsp1(C,k5_matrixc1(A,B)) ) ) ) ) ).
fof(t27_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m1_subset_1(C,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k5_matrixc1(k6_complsp1(C,A),B) = k6_complsp1(C,k5_matrixc1(A,B)) ) ) ) ) ).
fof(t28_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k1_complsp2(k5_matrixc1(A,B)) = k5_matrixc1(k1_complsp2(A),k1_complsp2(B)) ) ) ) ).
fof(t29_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m1_subset_1(B,k2_numbers)
=> k6_matrixc1(k6_complsp1(B,A)) = k5_binop_2(B,k6_matrixc1(A)) ) ) ).
fof(d9_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k10_matrixc1(A) = A ) ).
fof(t30_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( ( A = B
& r1_xreal_0(np__1,k3_finseq_1(A)) )
=> k2_finsop_1(k1_numbers,A,k33_binop_2) = k2_finsop_1(k2_numbers,B,k27_binop_2) ) ) ) ).
fof(t31_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( ( A = B
& r1_xreal_0(np__1,k3_finseq_1(A)) )
=> k15_rvsum_1(A) = k6_matrixc1(B) ) ) ) ).
fof(t32_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k6_matrixc1(k4_complsp1(A,B)) = k4_binop_2(k6_matrixc1(A),k6_matrixc1(B)) ) ) ) ).
fof(t33_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(k3_complsp1(A,B)))
=> k9_matrix_5(k3_complsp1(A,B),C) = k3_binop_2(k9_matrix_5(A,C),k9_matrix_5(B,C)) ) ) ) ) ).
fof(t34_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(k4_complsp1(A,B)))
=> k9_matrix_5(k4_complsp1(A,B),C) = k4_binop_2(k9_matrix_5(A,C),k9_matrix_5(B,C)) ) ) ) ) ).
fof(t35_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k5_matrixc1(k4_complsp1(A,B),C) = k4_complsp1(k5_matrixc1(A,C),k5_matrixc1(B,C)) ) ) ) ) ).
fof(t36_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k5_matrixc1(A,k4_complsp1(B,C)) = k4_complsp1(k5_matrixc1(A,B),k5_matrixc1(A,C)) ) ) ) ) ).
fof(t37_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k5_matrixc1(A,k3_complsp1(B,C)) = k3_complsp1(k5_matrixc1(A,B),k5_matrixc1(A,C)) ) ) ) ) ).
fof(t38_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k5_matrixc1(k3_complsp1(A,B),C) = k3_complsp1(k5_matrixc1(A,C),k5_matrixc1(B,C)) ) ) ) ) ).
fof(t39_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k6_matrixc1(k3_complsp1(A,B)) = k3_binop_2(k6_matrixc1(A),k6_matrixc1(B)) ) ) ) ).
fof(t40_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( ( A = C
& B = D
& k3_finseq_1(A) = k3_finseq_1(D) )
=> k1_finseqop(k2_numbers,k2_numbers,k2_numbers,k29_binop_2,A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k35_binop_2,C,D) ) ) ) ) ) ).
fof(t41_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> k10_matrixc1(k13_rvsum_1(A,B)) = k5_matrixc1(k10_matrixc1(A),k10_matrixc1(B)) ) ) ) ).
fof(t42_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| k8_complsp2(A,B) = k6_matrixc1(k5_matrixc1(A,k1_complsp2(B))) ) ) ) ) ).
fof(t43_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k1_matrix_1(A) = k1_matrix_1(B) )
=> k2_matrix_1(A) = k2_matrix_1(B) ) ) ) ).
fof(t44_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(C) = k3_finseq_1(D)
& k1_matrix_1(C) = k1_matrix_1(D)
& r2_hidden(B,k2_finseq_1(k3_finseq_1(C))) )
=> k7_matrix_1(k2_numbers,k3_matrix_5(C,D),B) = k3_complsp1(k7_matrix_1(k2_numbers,C,B),k7_matrix_1(k2_numbers,D,B)) ) ) ) ) ) ).
fof(t45_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( r2_hidden(A,k2_finseq_1(k3_finseq_1(B)))
=> k7_matrix_1(k2_numbers,B,A) = k1_complsp2(k7_matrix_1(k2_numbers,k1_matrixc1(B),A)) ) ) ) ).
fof(t46_matrixc1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( k3_finseq_1(B) = k1_matrix_1(C)
=> k5_matrixc1(B,k1_complsp2(k7_matrix_1(k2_numbers,k1_matrixc1(C),A))) = k1_complsp2(k5_matrixc1(k7_matrix_1(k2_numbers,k1_matrixc1(C),A),k1_complsp2(B))) ) ) ) ) ).
fof(t47_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( ( v1_matrix_1(B)
& m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k1_matrix_1(B)
& r1_xreal_0(np__1,k3_finseq_1(A)) )
=> k1_complsp2(k7_matrixc1(B,A)) = k7_matrixc1(k1_matrixc1(B),k1_complsp2(A)) ) ) ) ).
fof(t48_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_finseq_1(C,k2_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> k5_matrixc1(A,k5_matrixc1(B,C)) = k5_matrixc1(k5_matrixc1(A,B),C) ) ) ) ) ).
fof(t49_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> k6_matrixc1(k5_complsp1(A)) = k1_binop_2(k6_matrixc1(A)) ) ).
fof(t50_matrixc1,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> k6_matrixc1(k13_binarith(k2_numbers,A)) = A ) ).
fof(t51_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> k6_matrixc1(k8_finseq_1(k2_numbers,A,B)) = k3_binop_2(k6_matrixc1(A),k6_matrixc1(B)) ) ) ).
fof(d10_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( B = k11_matrixc1(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(k3_finseq_1(A)))
=> k9_matrix_5(B,C) = k6_matrixc1(k7_matrix_1(k2_numbers,A,C)) ) ) ) ) ) ) ).
fof(d11_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( B = k12_matrixc1(A)
<=> ( k3_finseq_1(B) = k1_matrix_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(k1_matrix_1(A)))
=> k9_matrix_5(B,C) = k6_matrixc1(k8_matrix_1(k2_numbers,A,C)) ) ) ) ) ) ) ).
fof(t52_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ( k3_finseq_1(A) = np__1
=> k6_matrixc1(A) = k9_matrix_5(A,np__1) ) ) ).
fof(t53_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( k3_finseq_1(A) = k23_binop_2(C,np__1)
& B = k16_finseq_1(k2_numbers,A,C) )
=> k6_matrixc1(A) = k3_binop_2(k6_matrixc1(B),k4_finseq_4(k5_numbers,k2_numbers,A,k3_finseq_1(A))) ) ) ) ) ).
fof(t54_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k6_matrixc1(k11_matrixc1(A)) = k6_matrixc1(k12_matrixc1(A)) ) ) ).
fof(d12_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k13_matrixc1(A) = k6_matrixc1(k11_matrixc1(A)) ) ).
fof(t55_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> k12_matrixc1(A) = k11_matrixc1(k4_matrix_1(k2_numbers,A)) ) ).
fof(t56_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
=> k13_matrixc1(A) = k13_matrixc1(k4_matrix_1(k2_numbers,A)) ) ) ).
fof(d13_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(C)
& k3_finseq_1(B) = k1_matrix_1(C) )
=> ! [D] :
( ( v1_matrix_1(D)
& m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
=> ( D = k14_matrixc1(A,B,C)
<=> ( k3_finseq_1(D) = k3_finseq_1(A)
& k1_matrix_1(D) = k3_finseq_1(B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(k4_tarski(E,F),k2_matrix_1(C))
=> k3_matrix_1(k2_numbers,D,E,F) = k5_binop_2(k5_binop_2(k9_matrix_5(A,E),k3_matrix_1(k2_numbers,C,E,F)),k15_complex1(k9_matrix_5(B,F))) ) ) ) ) ) ) ) ) ) ) ).
fof(t57_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(C)
& k3_finseq_1(B) = k1_matrix_1(C) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_finseq_1(B),np__0)
| k4_matrix_1(k2_numbers,k14_matrixc1(A,B,C)) = k1_matrixc1(k14_matrixc1(B,A,k2_matrixc1(C))) ) ) ) ) ) ).
fof(t58_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(C)
& k3_finseq_1(B) = k1_matrix_1(C) )
=> k1_matrixc1(k14_matrixc1(A,B,C)) = k14_matrixc1(k1_complsp2(A),k1_complsp2(B),k1_matrixc1(C)) ) ) ) ) ).
fof(t59_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(k3_finseq_1(B),np__0)
| k8_complsp2(A,B) = k15_complex1(k8_complsp2(B,A)) ) ) ) ) ).
fof(t60_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( r1_xreal_0(k3_finseq_1(B),np__0)
| k15_complex1(k8_complsp2(A,B)) = k8_complsp2(k1_complsp2(A),k1_complsp2(B)) ) ) ) ) ).
fof(t61_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( ~ r1_xreal_0(k1_matrix_1(A),np__0)
=> k1_matrixc1(k4_matrix_1(k2_numbers,A)) = k4_matrix_1(k2_numbers,k1_matrixc1(A)) ) ) ).
fof(t62_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k1_matrix_1(C)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_finseq_1(B),np__0)
| k8_complsp2(A,k7_matrixc1(k2_matrixc1(C),B)) = k13_matrixc1(k14_matrixc1(A,B,k4_matrix_1(k2_numbers,C))) ) ) ) ) ) ).
fof(t63_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(B) = k3_finseq_1(C)
& k3_finseq_1(A) = k1_matrix_1(C) )
=> ( r1_xreal_0(k3_finseq_1(A),np__0)
| r1_xreal_0(k3_finseq_1(B),np__0)
| r1_xreal_0(k3_finseq_1(C),np__0)
| k8_complsp2(k7_matrixc1(C,A),B) = k13_matrixc1(k14_matrixc1(A,B,k4_matrix_1(k2_numbers,C))) ) ) ) ) ) ).
fof(t64_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k1_matrix_1(C)
& k3_finseq_1(B) = k3_finseq_1(C) )
=> ( r1_xreal_0(k1_matrix_1(C),np__0)
| r1_xreal_0(k3_finseq_1(C),np__0)
| k8_complsp2(k7_matrixc1(C,A),B) = k8_complsp2(A,k7_matrixc1(k2_matrixc1(C),B)) ) ) ) ) ) ).
fof(t65_matrixc1,axiom,
! [A] :
( m2_finseq_1(A,k2_numbers)
=> ! [B] :
( m2_finseq_1(B,k2_numbers)
=> ! [C] :
( ( v1_matrix_1(C)
& m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( ( k3_finseq_1(A) = k3_finseq_1(C)
& k3_finseq_1(B) = k1_matrix_1(C) )
=> ( r1_xreal_0(k1_matrix_1(C),np__0)
| r1_xreal_0(k3_finseq_1(C),np__0)
| r1_xreal_0(k3_finseq_1(A),np__0)
| k8_complsp2(A,k7_matrixc1(C,B)) = k8_complsp2(k7_matrixc1(k2_matrixc1(C),A),B) ) ) ) ) ) ).
fof(dt_k1_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k1_matrixc1(A))
& m2_finseq_1(k1_matrixc1(A),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k2_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k2_matrixc1(A))
& m2_finseq_1(k2_matrixc1(A),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k3_matrixc1,axiom,
! [A] :
( m1_finseq_1(A,k2_numbers)
=> ( v1_matrix_1(k3_matrixc1(A))
& m2_finseq_1(k3_matrixc1(A),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k4_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> m2_finseq_1(k4_matrixc1(A),k2_numbers) ) ).
fof(dt_k5_matrixc1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k2_numbers)
& m1_finseq_1(B,k2_numbers) )
=> m2_finseq_1(k5_matrixc1(A,B),k2_numbers) ) ).
fof(commutativity_k5_matrixc1,axiom,
! [A,B] :
( ( m1_finseq_1(A,k2_numbers)
& m1_finseq_1(B,k2_numbers) )
=> k5_matrixc1(A,B) = k5_matrixc1(B,A) ) ).
fof(dt_k6_matrixc1,axiom,
! [A] :
( m1_finseq_1(A,k2_numbers)
=> m1_subset_1(k6_matrixc1(A),k2_numbers) ) ).
fof(dt_k7_matrixc1,axiom,
! [A,B] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers))
& m1_finseq_1(B,k2_numbers) )
=> m2_finseq_1(k7_matrixc1(A,B),k2_numbers) ) ).
fof(dt_k8_matrixc1,axiom,
! [A,B] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers))
& v1_xcmplx_0(B) )
=> ( v1_matrix_1(k8_matrixc1(A,B))
& m2_finseq_1(k8_matrixc1(A,B),k3_finseq_2(k2_numbers)) ) ) ).
fof(dt_k9_matrixc1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> m2_finseq_2(k9_matrixc1(A,B,C),k2_numbers,k4_finseq_2(A,k2_numbers)) ) ).
fof(commutativity_k9_matrixc1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> k9_matrixc1(A,B,C) = k9_matrixc1(A,C,B) ) ).
fof(redefinition_k9_matrixc1,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k2_numbers))
& m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
=> k9_matrixc1(A,B,C) = k5_matrixc1(B,C) ) ).
fof(dt_k10_matrixc1,axiom,
! [A] :
( m1_finseq_1(A,k1_numbers)
=> m2_finseq_1(k10_matrixc1(A),k2_numbers) ) ).
fof(dt_k11_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> m2_finseq_1(k11_matrixc1(A),k2_numbers) ) ).
fof(dt_k12_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> m2_finseq_1(k12_matrixc1(A),k2_numbers) ) ).
fof(dt_k13_matrixc1,axiom,
! [A] :
( ( v1_matrix_1(A)
& m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
=> m1_subset_1(k13_matrixc1(A),k2_numbers) ) ).
fof(dt_k14_matrixc1,axiom,
! [A,B,C] :
( ( m1_finseq_1(A,k2_numbers)
& m1_finseq_1(B,k2_numbers)
& v1_matrix_1(C)
& m1_finseq_1(C,k3_finseq_2(k2_numbers)) )
=> ( v1_matrix_1(k14_matrixc1(A,B,C))
& m2_finseq_1(k14_matrixc1(A,B,C),k3_finseq_2(k2_numbers)) ) ) ).
%------------------------------------------------------------------------------