SET007 Axioms: SET007+574.ax
%------------------------------------------------------------------------------
% File : SET007+574 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Construction of SCM over Ring
% 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 : scmring1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 70 ( 0 unt; 0 def)
% Number of atoms : 551 ( 99 equ)
% Maximal formula atoms : 61 ( 7 avg)
% Number of connectives : 571 ( 90 ~; 0 |; 294 &)
% ( 17 <=>; 170 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 9 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of predicates : 44 ( 43 usr; 0 prp; 1-3 aty)
% Number of functors : 67 ( 67 usr; 18 con; 0-6 aty)
% Number of variables : 230 ( 175 !; 55 ?)
% 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(cc1_scmring1,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ( ~ v1_xboole_0(A)
& ~ v1_realset1(A) ) ) ).
fof(cc2_scmring1,axiom,
! [A] :
( l1_struct_0(A)
=> ( ~ v6_group_1(A)
=> ( ~ v3_struct_0(A)
& ~ v3_realset2(A) ) ) ) ).
fof(cc3_scmring1,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A) ) ) ) ).
fof(cc4_scmring1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v6_vectsp_1(A) ) ) ) ).
fof(cc5_scmring1,axiom,
! [A] :
( m1_subset_1(A,k2_ami_2)
=> ( v4_ordinal2(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xcmplx_0(A) ) ) ).
fof(fc1_scmring1,axiom,
( v1_relat_1(k4_ami_2)
& ~ v1_xboole_0(k4_ami_2)
& ~ v1_realset1(k4_ami_2) ) ).
fof(fc2_scmring1,axiom,
( ~ v1_xboole_0(k3_ami_2)
& ~ v1_finset_1(k3_ami_2)
& ~ v1_realset1(k3_ami_2)
& v1_membered(k3_ami_2)
& v2_membered(k3_ami_2)
& v3_membered(k3_ami_2)
& v4_membered(k3_ami_2)
& v5_membered(k3_ami_2) ) ).
fof(fc3_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_relat_1(k1_scmring1(A))
& ~ v1_xboole_0(k1_scmring1(A))
& ~ v1_realset1(k1_scmring1(A)) ) ) ).
fof(cc6_scmring1,axiom,
! [A] :
( l1_struct_0(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v1_scmring1(A) ) ) ) ).
fof(rc1_scmring1,axiom,
? [A] :
( l1_struct_0(A)
& v1_struct_0(A)
& ~ v3_struct_0(A)
& v6_group_1(A)
& v3_realset2(A)
& v1_scmring1(A) ) ).
fof(rc2_scmring1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v3_realset2(A)
& v1_scmring1(A) ) ).
fof(rc3_scmring1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& v6_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v3_realset2(A)
& v1_scmring1(A) ) ).
fof(fc4_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
& m1_relset_1(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
& m1_subset_1(C,k1_scmring1(A)) )
=> ( v1_relat_1(k1_funct_1(B,C))
& v1_funct_1(k1_funct_1(B,C)) ) ) ).
fof(d2_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_scmring1(A)
<=> ( u1_struct_0(A) != k3_ami_2
& u1_struct_0(A) != k1_scmring1(A) ) ) ) ).
fof(d3_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)))
& m2_relset_1(B,k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2))) )
=> ( B = k2_scmring1(A)
<=> ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)),B,np__0) = k3_ami_2
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)),B,k1_nat_1(k2_nat_1(np__2,C),np__1)) = u1_struct_0(A)
& k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)),B,k1_nat_1(k2_nat_1(np__2,C),np__2)) = k1_scmring1(A) ) ) ) ) ) ) ).
fof(t1_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> k3_ami_2 != k1_scmring1(A) ) ).
fof(t2_scmring1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = k3_ami_2
<=> A = np__0 ) ) ) ).
fof(t3_scmring1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = u1_struct_0(B)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_nat_1(k2_nat_1(np__2,C),np__1) ) ) ) ) ).
fof(t4_scmring1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> ( k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = k1_scmring1(B)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k1_nat_1(k2_nat_1(np__2,C),np__2) ) ) ) ) ).
fof(t5_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k2_ami_2)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = u1_struct_0(B) ) ) ).
fof(t6_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k3_ami_2)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> k8_funct_2(k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(B)),k2_tarski(k1_scmring1(B),k3_ami_2)),k2_scmring1(B),A) = k1_scmring1(B) ) ) ).
fof(t7_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> k5_card_3(np__0,k4_card_3(k2_scmring1(A))) = k3_ami_2 ) ).
fof(t8_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k2_ami_2)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> k5_card_3(A,k4_card_3(k2_scmring1(B))) = u1_struct_0(B) ) ) ).
fof(t9_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k3_ami_2)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_scmring1(B)
& l1_struct_0(B) )
=> k5_card_3(A,k4_card_3(k2_scmring1(B))) = k1_scmring1(B) ) ) ).
fof(d4_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> k3_scmring1(A,B) = k1_funct_1(B,np__0) ) ) ).
fof(d5_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
=> k4_scmring1(A,B,C) = k1_funct_4(B,k3_cqc_lang(np__0,C)) ) ) ) ).
fof(t10_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
=> k1_funct_1(k4_scmring1(A,B,C),np__0) = C ) ) ) ).
fof(t11_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> k1_funct_1(k4_scmring1(A,B,C),D) = k1_funct_1(B,D) ) ) ) ) ).
fof(t12_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
=> ! [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
=> k1_funct_1(k4_scmring1(A,B,C),D) = k1_funct_1(B,D) ) ) ) ) ).
fof(d6_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k5_scmring1(A,B,C,D) = k1_funct_4(B,k3_cqc_lang(C,D)) ) ) ) ) ).
fof(t13_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_funct_1(k5_scmring1(A,B,C,D),np__0) = k1_funct_1(B,np__0) ) ) ) ) ).
fof(t14_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_funct_1(k5_scmring1(A,B,C,D),C) = D ) ) ) ) ).
fof(t15_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> ( E != C
=> k1_funct_1(k5_scmring1(A,B,C,D),E) = k1_funct_1(B,E) ) ) ) ) ) ) ).
fof(t16_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k4_card_3(k2_scmring1(A)))
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m2_subset_1(E,k5_numbers,k3_ami_2)
=> k1_funct_1(k5_scmring1(A,B,C,D),E) = k1_funct_1(B,E) ) ) ) ) ) ).
fof(d7_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
& ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(E,k2_finseq_4(k2_ami_2,C,D)) ) ) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ( C = k7_scmring1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k2_ami_2)
& D = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k2_ami_2,D,np__1) ) ) ) ) ) ) ).
fof(d8_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
& ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(E,k2_finseq_4(k2_ami_2,C,D)) ) ) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ( C = k8_scmring1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k2_ami_2)
& D = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k2_ami_2,D,np__2) ) ) ) ) ) ) ).
fof(t17_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,k1_scmring1(B))
=> ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> ( C = k4_tarski(A,k2_finseq_4(k2_ami_2,D,E))
=> ( k7_scmring1(B,C) = D
& k8_scmring1(B,C) = E ) ) ) ) ) ) ) ).
fof(d9_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
& ? [D] :
( m2_subset_1(D,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(D,k12_finseq_1(k3_ami_2,C)) ) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
=> ( C = k9_scmring1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k3_ami_2)
& D = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k3_ami_2,D,np__1) ) ) ) ) ) ) ).
fof(t18_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,k1_scmring1(B))
=> ! [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
=> ( C = k4_tarski(A,k12_finseq_1(k3_ami_2,D))
=> k9_scmring1(B,C) = D ) ) ) ) ) ).
fof(d10_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
& ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(E,k2_finseq_4(k5_numbers,C,D)) ) ) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
=> ( C = k10_scmring1(A,B)
<=> ? [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& k2_finseq_4(k5_numbers,D,E) = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k5_numbers,k2_finseq_4(k5_numbers,D,E),np__1) ) ) ) ) ) ) ) ).
fof(d11_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k3_ami_2)
& ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(E,k2_finseq_4(k5_numbers,C,D)) ) ) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ( C = k11_scmring1(A,B)
<=> ? [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& k2_finseq_4(k5_numbers,D,E) = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k5_numbers,k2_finseq_4(k5_numbers,D,E),np__2) ) ) ) ) ) ) ) ).
fof(t19_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,k1_scmring1(B))
=> ! [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> ( C = k4_tarski(A,k2_finseq_4(k5_numbers,D,E))
=> ( k10_scmring1(B,C) = D
& k11_scmring1(B,C) = E ) ) ) ) ) ) ) ).
fof(d12_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(E,k12_scmring1(A,C,D)) ) ) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
=> ( C = k13_scmring1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k2_xboole_0(k2_ami_2,u1_struct_0(A)))
& D = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k2_xboole_0(k2_ami_2,u1_struct_0(A)),D,np__1) ) ) ) ) ) ) ).
fof(d13_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ( ? [C] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
& ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m2_subset_1(E,k5_numbers,k1_gr_cy_1(np__8))
& B = k4_tarski(E,k12_scmring1(A,C,D)) ) ) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k14_scmring1(A,B)
<=> ? [D] :
( m2_finseq_1(D,k2_xboole_0(k2_ami_2,u1_struct_0(A)))
& D = k2_mcart_1(B)
& C = k4_finseq_4(k5_numbers,k2_xboole_0(k2_ami_2,u1_struct_0(A)),D,np__2) ) ) ) ) ) ) ).
fof(t20_scmring1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_gr_cy_1(np__8))
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,k1_scmring1(B))
=> ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( C = k4_tarski(A,k12_scmring1(B,D,E))
=> ( k13_scmring1(B,C) = D
& k14_scmring1(B,C) = E ) ) ) ) ) ) ) ).
fof(d14_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_scmring1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_scmring1(A))
=> ! [C] :
( m1_subset_1(C,k4_card_3(k2_scmring1(A)))
=> ( ( ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& B = k4_tarski(np__1,k2_finseq_4(k2_ami_2,D,E)) ) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k6_scmring1(A,C,k8_scmring1(A,B))),k15_ami_2(k3_scmring1(A,C))) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& B = k4_tarski(np__2,k2_finseq_4(k2_ami_2,D,E)) ) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k4_rlvect_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& B = k4_tarski(np__3,k2_finseq_4(k2_ami_2,D,E)) ) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k6_rlvect_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& B = k4_tarski(np__4,k2_finseq_4(k2_ami_2,D,E)) ) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k1_group_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
& B = k4_tarski(np__6,k12_finseq_1(k3_ami_2,D)) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,C,k9_scmring1(A,B)) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
& ? [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
& B = k4_tarski(np__7,k2_finseq_4(k5_numbers,D,E)) ) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,C,k2_cqc_lang(k3_ami_2,k6_scmring1(A,C,k11_scmring1(A,B)),k1_rlvect_1(A),k10_scmring1(A,B),k15_ami_2(k3_scmring1(A,C)))) )
& ( ? [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& B = k4_tarski(np__5,k12_scmring1(A,D,E)) ) )
=> k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k13_scmring1(A,B),k14_scmring1(A,B)),k15_ami_2(k3_scmring1(A,C))) )
& ( ( ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> B != k4_tarski(np__1,k2_finseq_4(k2_ami_2,D,E)) ) )
& ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> B != k4_tarski(np__2,k2_finseq_4(k2_ami_2,D,E)) ) )
& ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> B != k4_tarski(np__3,k2_finseq_4(k2_ami_2,D,E)) ) )
& ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> B != k4_tarski(np__4,k2_finseq_4(k2_ami_2,D,E)) ) )
& ! [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
=> B != k4_tarski(np__6,k12_finseq_1(k3_ami_2,D)) )
& ! [D] :
( m2_subset_1(D,k5_numbers,k3_ami_2)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k2_ami_2)
=> B != k4_tarski(np__7,k2_finseq_4(k5_numbers,D,E)) ) )
& ! [D] :
( m2_subset_1(D,k5_numbers,k2_ami_2)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> B != k4_tarski(np__5,k12_scmring1(A,D,E)) ) ) )
=> k15_scmring1(A,B,C) = C ) ) ) ) ) ).
fof(d15_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_scmring1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
& m2_relset_1(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) )
=> ( B = k16_scmring1(A)
<=> ! [C] :
( m1_subset_1(C,k1_scmring1(A))
=> ! [D] :
( m1_subset_1(D,k4_card_3(k2_scmring1(A)))
=> k8_funct_2(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)),k1_cat_2(k1_scmring1(A),k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))),B,C),D) = k15_scmring1(A,C,D) ) ) ) ) ) ).
fof(dt_k1_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> m1_subset_1(k1_scmring1(A),k1_zfmisc_1(k2_zfmisc_1(k1_gr_cy_1(np__8),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),k5_numbers))))) ) ).
fof(dt_k2_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ( v1_funct_1(k2_scmring1(A))
& v1_funct_2(k2_scmring1(A),k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2)))
& m2_relset_1(k2_scmring1(A),k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2))) ) ) ).
fof(dt_k3_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k4_card_3(k2_scmring1(A))) )
=> m2_subset_1(k3_scmring1(A,B),k5_numbers,k3_ami_2) ) ).
fof(dt_k4_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A)
& m1_subset_1(B,k4_card_3(k2_scmring1(A)))
& m1_subset_1(C,k3_ami_2) )
=> m1_subset_1(k4_scmring1(A,B,C),k4_card_3(k2_scmring1(A))) ) ).
fof(dt_k5_scmring1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A)
& m1_subset_1(B,k4_card_3(k2_scmring1(A)))
& m1_subset_1(C,k2_ami_2)
& m1_subset_1(D,u1_struct_0(A)) )
=> m1_subset_1(k5_scmring1(A,B,C,D),k4_card_3(k2_scmring1(A))) ) ).
fof(dt_k6_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A)
& m1_subset_1(B,k4_card_3(k2_scmring1(A)))
& m1_subset_1(C,k2_ami_2) )
=> m1_subset_1(k6_scmring1(A,B,C),u1_struct_0(A)) ) ).
fof(redefinition_k6_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v1_scmring1(A)
& l1_struct_0(A)
& m1_subset_1(B,k4_card_3(k2_scmring1(A)))
& m1_subset_1(C,k2_ami_2) )
=> k6_scmring1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k7_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m2_subset_1(k7_scmring1(A,B),k5_numbers,k2_ami_2) ) ).
fof(dt_k8_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m2_subset_1(k8_scmring1(A,B),k5_numbers,k2_ami_2) ) ).
fof(dt_k9_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m2_subset_1(k9_scmring1(A,B),k5_numbers,k3_ami_2) ) ).
fof(dt_k10_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m2_subset_1(k10_scmring1(A,B),k5_numbers,k3_ami_2) ) ).
fof(dt_k11_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m2_subset_1(k11_scmring1(A,B),k5_numbers,k2_ami_2) ) ).
fof(dt_k12_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k2_ami_2)
& m1_subset_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k12_scmring1(A,B,C),k2_xboole_0(k2_ami_2,u1_struct_0(A))) ) ).
fof(redefinition_k12_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k2_ami_2)
& m1_subset_1(C,u1_struct_0(A)) )
=> k12_scmring1(A,B,C) = k10_finseq_1(B,C) ) ).
fof(dt_k13_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m2_subset_1(k13_scmring1(A,B),k5_numbers,k2_ami_2) ) ).
fof(dt_k14_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_scmring1(A)) )
=> m1_subset_1(k14_scmring1(A,B),u1_struct_0(A)) ) ).
fof(dt_k15_scmring1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_scmring1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_scmring1(A))
& m1_subset_1(C,k4_card_3(k2_scmring1(A))) )
=> m1_subset_1(k15_scmring1(A,B,C),k4_card_3(k2_scmring1(A))) ) ).
fof(dt_k16_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_scmring1(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k16_scmring1(A))
& v1_funct_2(k16_scmring1(A),k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A))))
& m2_relset_1(k16_scmring1(A),k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) ) ) ).
fof(d1_scmring1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> k1_scmring1(A) = k2_xboole_0(k2_xboole_0(k2_xboole_0(k2_xboole_0(k1_tarski(k4_tarski(np__0,k1_xboole_0)),a_0_0_scmring1),a_0_1_scmring1),a_0_2_scmring1),a_1_0_scmring1(A)) ) ).
fof(fraenkel_a_0_0_scmring1,axiom,
! [A] :
( r2_hidden(A,a_0_0_scmring1)
<=> ? [B,C,D] :
( m2_subset_1(B,k5_numbers,k1_gr_cy_1(np__8))
& m2_subset_1(C,k5_numbers,k2_ami_2)
& m2_subset_1(D,k5_numbers,k2_ami_2)
& A = k4_tarski(B,k2_finseq_4(k2_ami_2,C,D))
& r2_hidden(B,k2_enumset1(np__1,np__2,np__3,np__4)) ) ) ).
fof(fraenkel_a_0_1_scmring1,axiom,
! [A] :
( r2_hidden(A,a_0_1_scmring1)
<=> ? [B] :
( m2_subset_1(B,k5_numbers,k3_ami_2)
& A = k4_tarski(np__6,k12_finseq_1(k3_ami_2,B)) ) ) ).
fof(fraenkel_a_0_2_scmring1,axiom,
! [A] :
( r2_hidden(A,a_0_2_scmring1)
<=> ? [B,C] :
( m2_subset_1(B,k5_numbers,k3_ami_2)
& m2_subset_1(C,k5_numbers,k2_ami_2)
& A = k4_tarski(np__7,k2_finseq_4(k5_numbers,B,C)) ) ) ).
fof(fraenkel_a_1_0_scmring1,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B) )
=> ( r2_hidden(A,a_1_0_scmring1(B))
<=> ? [C,D] :
( m2_subset_1(C,k5_numbers,k2_ami_2)
& m1_subset_1(D,u1_struct_0(B))
& A = k4_tarski(np__5,k10_finseq_1(C,D)) ) ) ) ).
%------------------------------------------------------------------------------