SET007 Axioms: SET007+678.ax
%------------------------------------------------------------------------------
% File : SET007+678 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Ring Ideals
% 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 : ideal_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 226 ( 0 unt; 0 def)
% Number of atoms : 2160 ( 96 equ)
% Maximal formula atoms : 31 ( 9 avg)
% Number of connectives : 2414 ( 480 ~; 1 |;1378 &)
% ( 37 <=>; 518 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of predicates : 69 ( 68 usr; 0 prp; 1-4 aty)
% Number of functors : 79 ( 79 usr; 5 con; 0-5 aty)
% Number of variables : 686 ( 636 !; 50 ?)
% 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_ideal_1,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v1_algstr_1(A) ) ).
fof(rc2_ideal_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& ~ v3_realset2(A) ) ).
fof(rc3_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A) ) ) ).
fof(rc4_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A) ) ) ).
fof(rc5_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v3_ideal_1(B,A) ) ) ).
fof(rc6_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A) ) ) ).
fof(rc7_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A) ) ) ).
fof(rc8_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A) ) ) ).
fof(cc1_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v1_xboole_0(B)
& v2_ideal_1(B,A) )
=> ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A) ) ) ) ) ).
fof(cc2_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A) )
=> ( ~ v1_xboole_0(B)
& v2_ideal_1(B,A) ) ) ) ) ).
fof(fc1_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> v1_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) ).
fof(fc2_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> v2_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) ).
fof(fc3_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> v3_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) ).
fof(rc9_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v5_ideal_1(B,A) ) ) ).
fof(rc10_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& v5_ideal_1(B,A) ) ) ).
fof(fc4_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k3_ideal_1(A,B))
& v4_rlvect_1(k3_ideal_1(A,B)) ) ) ).
fof(fc5_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k3_ideal_1(A,B))
& v4_rlvect_1(k3_ideal_1(A,B))
& v5_rlvect_1(k3_ideal_1(A,B)) ) ) ).
fof(fc6_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k3_ideal_1(A,B))
& v3_rlvect_1(k3_ideal_1(A,B)) ) ) ).
fof(fc7_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(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)
& v1_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k3_ideal_1(A,B))
& v3_rlvect_1(k3_ideal_1(A,B))
& v4_rlvect_1(k3_ideal_1(A,B))
& v5_rlvect_1(k3_ideal_1(A,B))
& v6_rlvect_1(k3_ideal_1(A,B))
& v1_binom(k3_ideal_1(A,B))
& v1_algstr_1(k3_ideal_1(A,B))
& v2_algstr_1(k3_ideal_1(A,B))
& v3_algstr_1(k3_ideal_1(A,B))
& v4_algstr_1(k3_ideal_1(A,B))
& v5_algstr_1(k3_ideal_1(A,B))
& v6_algstr_1(k3_ideal_1(A,B)) ) ) ).
fof(rc11_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m1_ideal_1(C,A,B)
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C)
& v1_finseq_1(C)
& v1_polynom1(C) ) ) ).
fof(rc12_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m2_ideal_1(C,A,B)
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C)
& v1_finseq_1(C)
& v1_polynom1(C) ) ) ).
fof(rc13_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( m3_ideal_1(C,A,B)
& ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C)
& v1_finseq_1(C)
& v1_polynom1(C) ) ) ).
fof(fc8_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> ~ v1_xboole_0(k10_ideal_1(A,B,C)) ) ).
fof(fc9_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v1_ideal_1(k10_ideal_1(A,B,C),A) ) ).
fof(fc10_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l3_vectsp_1(A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> v3_ideal_1(k10_ideal_1(A,B,C),A) ) ).
fof(fc11_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k11_ideal_1(A,B,C)) ) ).
fof(fc12_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& l1_rlvect_1(A)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_ideal_1(k11_ideal_1(A,B,C),A) ) ).
fof(fc13_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v3_ideal_1(k11_ideal_1(A,B,C),A) ) ).
fof(fc14_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v2_ideal_1(k11_ideal_1(A,B,C),A) ) ).
fof(fc15_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k3_xboole_0(B,C)) ) ).
fof(fc16_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v1_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v1_ideal_1(k3_xboole_0(B,C),A) ) ).
fof(fc17_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v2_ideal_1(k3_xboole_0(B,C),A) ) ).
fof(fc18_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> v3_ideal_1(k3_xboole_0(B,C),A) ) ).
fof(fc19_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k14_ideal_1(A,B,C)) ) ).
fof(fc20_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k14_ideal_1(A,B,C))
& v1_ideal_1(k14_ideal_1(A,B,C),A) ) ) ).
fof(fc21_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k14_ideal_1(A,B,C))
& v3_ideal_1(k14_ideal_1(A,B,C),A) ) ) ).
fof(fc22_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k14_ideal_1(A,B,C))
& v2_ideal_1(k14_ideal_1(A,B,C),A) ) ) ).
fof(fc23_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k16_ideal_1(A,B,C)) ) ).
fof(fc24_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k16_ideal_1(A,B,C))
& v1_ideal_1(k16_ideal_1(A,B,C),A) ) ) ).
fof(fc25_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k16_ideal_1(A,B,C))
& v2_ideal_1(k16_ideal_1(A,B,C),A)
& v3_ideal_1(k16_ideal_1(A,B,C),A) ) ) ).
fof(fc26_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k16_ideal_1(A,B,C))
& v2_ideal_1(k16_ideal_1(A,B,C),A)
& v3_ideal_1(k16_ideal_1(A,B,C),A) ) ) ).
fof(fc27_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v1_xboole_0(k17_ideal_1(A,B)) ) ).
fof(fc28_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k17_ideal_1(A,B))
& v1_ideal_1(k17_ideal_1(A,B),A) ) ) ).
fof(fc29_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k17_ideal_1(A,B))
& v2_ideal_1(k17_ideal_1(A,B),A)
& v3_ideal_1(k17_ideal_1(A,B),A) ) ) ).
fof(fc30_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k17_ideal_1(A,B))
& v2_ideal_1(k17_ideal_1(A,B),A)
& v3_ideal_1(k17_ideal_1(A,B),A) ) ) ).
fof(rc14_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& v6_ideal_1(B,A) ) ) ).
fof(fc31_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k7_ideal_1(A,B))
& v1_ideal_1(k7_ideal_1(A,B),A)
& v2_ideal_1(k7_ideal_1(A,B),A)
& v3_ideal_1(k7_ideal_1(A,B),A)
& v6_ideal_1(k7_ideal_1(A,B),A) ) ) ).
fof(rc15_ideal_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_int_3(A)
& v1_binom(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A)
& v6_algstr_1(A)
& ~ v3_realset2(A) ) ).
fof(fc32_ideal_1,axiom,
( ~ v3_struct_0(k1_int_3)
& v2_group_1(k1_int_3)
& v4_group_1(k1_int_3)
& v7_group_1(k1_int_3)
& v3_rlvect_1(k1_int_3)
& v4_rlvect_1(k1_int_3)
& v5_rlvect_1(k1_int_3)
& v6_rlvect_1(k1_int_3)
& v3_vectsp_1(k1_int_3)
& v4_vectsp_1(k1_int_3)
& v5_vectsp_1(k1_int_3)
& v6_vectsp_1(k1_int_3)
& v7_vectsp_1(k1_int_3)
& v8_vectsp_1(k1_int_3)
& ~ v10_vectsp_1(k1_int_3)
& v1_int_3(k1_int_3)
& v1_binom(k1_int_3)
& v2_vectsp_2(k1_int_3)
& v3_gcd_1(k1_int_3)
& v1_algstr_1(k1_int_3)
& v2_algstr_1(k1_int_3)
& v3_algstr_1(k1_int_3)
& v4_algstr_1(k1_int_3)
& v5_algstr_1(k1_int_3)
& v6_algstr_1(k1_int_3)
& ~ v3_realset2(k1_int_3)
& v7_ideal_1(k1_int_3) ) ).
fof(rc16_ideal_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A)
& v6_algstr_1(A)
& ~ v3_realset2(A)
& v7_ideal_1(A) ) ).
fof(fc33_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ( ~ v1_xboole_0(k7_relat_1(C,D))
& v1_relat_1(k7_relat_1(C,D)) ) ) ).
fof(t1_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,C)) = k2_rlvect_1(A,B,C) ) ) ) ).
fof(d1_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_ideal_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k2_rlvect_1(A,C,D),B) ) ) ) ) ) ) ).
fof(d2_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_ideal_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> r2_hidden(k1_group_1(A,C,D),B) ) ) ) ) ) ) ).
fof(d3_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v3_ideal_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,B)
=> r2_hidden(k1_group_1(A,D,C),B) ) ) ) ) ) ) ).
fof(t2_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k1_rlvect_1(A),B) ) ) ).
fof(t3_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> r2_hidden(k1_rlvect_1(A),B) ) ) ).
fof(t4_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A) ) ).
fof(t5_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> v2_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A) ) ).
fof(t6_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> v3_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A) ) ).
fof(t7_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(k1_struct_0(A,k1_rlvect_1(A)))
& v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& v2_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& v3_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& m1_subset_1(k1_struct_0(A,k1_rlvect_1(A)),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t8_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(k1_struct_0(A,k1_rlvect_1(A)))
& v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& v2_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& m1_subset_1(k1_struct_0(A,k1_rlvect_1(A)),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t9_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(k1_struct_0(A,k1_rlvect_1(A)))
& v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& v3_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
& m1_subset_1(k1_struct_0(A,k1_rlvect_1(A)),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t10_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(u1_struct_0(A))
& v1_ideal_1(u1_struct_0(A),A)
& v2_ideal_1(u1_struct_0(A),A)
& v3_ideal_1(u1_struct_0(A),A)
& m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t11_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(u1_struct_0(A))
& v1_ideal_1(u1_struct_0(A),A)
& v2_ideal_1(u1_struct_0(A),A)
& m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(t12_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(u1_struct_0(A))
& v1_ideal_1(u1_struct_0(A),A)
& v3_ideal_1(u1_struct_0(A),A)
& m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(d4_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_realset1(B)
<=> B = k1_struct_0(A,k1_rlvect_1(A)) ) ) ) ).
fof(d5_ideal_1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v5_ideal_1(B,A)
<=> B != u1_struct_0(A) ) ) ) ).
fof(t13_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k5_rlvect_1(A,C),B) ) ) ) ) ).
fof(t14_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v6_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,B)
=> r2_hidden(k5_rlvect_1(A,C),B) ) ) ) ) ).
fof(t15_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k6_rlvect_1(A,C,D),B) ) ) ) ) ) ).
fof(t16_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v6_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k6_rlvect_1(A,C,D),B) ) ) ) ) ) ).
fof(t17_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> r2_hidden(k5_binom(A,C,D),B) ) ) ) ) ).
fof(t18_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_struct_0(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D != np__0
=> r2_hidden(k2_binom(A,C,D),B) ) ) ) ) ) ).
fof(d6_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k1_ideal_1(A,B) = k1_realset1(u1_rlvect_1(A),B) ) ) ).
fof(d7_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k2_ideal_1(A,B) = k1_realset1(u1_group_1(A),B) ) ) ).
fof(d8_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k3_ideal_1(A,B) = g1_rlvect_1(B,k1_ideal_1(A,B),k1_funct_7(k1_rlvect_1(A),B)) ) ) ).
fof(t19_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v5_ideal_1(B,A)
<=> ~ r2_hidden(k2_group_1(A),B) ) ) ) ).
fof(t20_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v5_ideal_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( v1_gcd_1(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(t21_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v5_ideal_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( v1_gcd_1(C,A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(t22_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_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)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( B = k1_struct_0(A,k1_rlvect_1(A))
| B = u1_struct_0(A) ) ) ) ) ).
fof(d9_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ( m1_ideal_1(C,A,B)
<=> ! [D] :
~ ( r2_hidden(D,k4_relset_1(k5_numbers,u1_struct_0(A),C))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_struct_0(G,A,B)
=> k4_finseq_4(k5_numbers,u1_struct_0(A),C,D) != k1_group_1(A,k1_group_1(A,E,G),F) ) ) ) ) ) ) ) ) ).
fof(d10_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ( m2_ideal_1(C,A,B)
<=> ! [D] :
~ ( r2_hidden(D,k4_relset_1(k5_numbers,u1_struct_0(A),C))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_struct_0(F,A,B)
=> k4_finseq_4(k5_numbers,u1_struct_0(A),C,D) != k1_group_1(A,E,F) ) ) ) ) ) ) ) ).
fof(d11_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(A))
=> ( m3_ideal_1(C,A,B)
<=> ! [D] :
~ ( r2_hidden(D,k4_relset_1(k5_numbers,u1_struct_0(A),C))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_struct_0(F,A,B)
=> k4_finseq_4(k5_numbers,u1_struct_0(A),C,D) != k1_group_1(A,F,E) ) ) ) ) ) ) ) ).
fof(t23_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_ideal_1(D,A,B)
=> m1_ideal_1(k6_polynom1(A,D,C),A,B) ) ) ) ) ).
fof(t24_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_ideal_1(D,A,B)
=> m1_ideal_1(k7_polynom1(A,D,C),A,B) ) ) ) ) ).
fof(t25_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> m1_ideal_1(C,A,B) ) ) ) ).
fof(t26_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_ideal_1(D,A,B)
=> m2_ideal_1(k6_polynom1(A,D,C),A,B) ) ) ) ) ).
fof(t27_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_ideal_1(D,A,B)
=> m1_ideal_1(k7_polynom1(A,D,C),A,B) ) ) ) ) ).
fof(t28_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m3_ideal_1(C,A,B)
=> m1_ideal_1(C,A,B) ) ) ) ).
fof(t29_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m3_ideal_1(D,A,B)
=> m3_ideal_1(k7_polynom1(A,D,C),A,B) ) ) ) ) ).
fof(t30_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m3_ideal_1(D,A,B)
=> m1_ideal_1(k6_polynom1(A,D,C),A,B) ) ) ) ) ).
fof(t31_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_ideal_1(C,A,B)
=> ( m2_ideal_1(C,A,B)
& m3_ideal_1(C,A,B) ) ) ) ) ).
fof(t32_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_ideal_1(C,A,B) )
=> ? [D] :
( m1_ideal_1(D,A,B)
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& C = k8_finseq_1(u1_struct_0(A),D,k12_finseq_1(u1_struct_0(A),E))
& m1_ideal_1(k12_finseq_1(u1_struct_0(A),E),A,B) ) ) ) ) ) ).
fof(t33_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_ideal_1(C,A,B) )
=> ? [D] :
( m2_ideal_1(D,A,B)
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& C = k8_finseq_1(u1_struct_0(A),D,k12_finseq_1(u1_struct_0(A),E))
& m2_ideal_1(k12_finseq_1(u1_struct_0(A),E),A,B) ) ) ) ) ) ).
fof(t34_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m3_ideal_1(C,A,B) )
=> ? [D] :
( m3_ideal_1(D,A,B)
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& C = k8_finseq_1(u1_struct_0(A),D,k12_finseq_1(u1_struct_0(A),E))
& m3_ideal_1(k12_finseq_1(u1_struct_0(A),E),A,B) ) ) ) ) ) ).
fof(d12_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_ideal_1(C,A,B)
=> ! [D] :
( m2_finseq_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
=> ( r1_ideal_1(A,B,C,D)
<=> ( k3_finseq_1(D) = k3_finseq_1(C)
& ! [E] :
( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),C))
=> ( k1_funct_1(C,E) = k1_group_1(A,k1_group_1(A,k5_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E)),k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E))),k7_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E)))
& r2_hidden(k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E)),B) ) ) ) ) ) ) ) ) ).
fof(t35_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_ideal_1(C,A,B)
=> ? [D] :
( m2_finseq_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
& r1_ideal_1(A,B,C,D) ) ) ) ) ).
fof(t36_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m1_ideal_1(D,A,C)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
=> ! [G] :
( m2_finseq_1(G,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
=> ~ ( r1_tarski(k4_pre_topc(A,B,F,C),E)
& r1_ideal_1(A,C,D,G)
& ! [H] :
( m1_ideal_1(H,B,E)
=> ~ ( k3_finseq_1(D) = k3_finseq_1(H)
& ! [I] :
( r2_hidden(I,k4_relset_1(k5_numbers,u1_struct_0(B),H))
=> k1_funct_1(H,I) = k1_group_1(B,k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k5_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),G,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),G,I)))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k7_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),G,I)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d13_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> ! [D] :
( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> ( r2_ideal_1(A,B,C,D)
<=> ( k3_finseq_1(D) = k3_finseq_1(C)
& ! [E] :
( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),C))
=> ( k1_funct_1(C,E) = k1_group_1(A,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)))
& r2_hidden(k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),B) ) ) ) ) ) ) ) ) ).
fof(t37_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> ? [D] :
( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
& r2_ideal_1(A,B,C,D) ) ) ) ) ).
fof(t38_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m2_ideal_1(D,A,C)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
=> ! [G] :
( m2_finseq_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> ~ ( r1_tarski(k4_pre_topc(A,B,F,C),E)
& r2_ideal_1(A,C,D,G)
& ! [H] :
( m2_ideal_1(H,B,E)
=> ~ ( k3_finseq_1(D) = k3_finseq_1(H)
& ! [I] :
( r2_hidden(I,k4_relset_1(k5_numbers,u1_struct_0(B),H))
=> k1_funct_1(H,I) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d14_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m3_ideal_1(C,A,B)
=> ! [D] :
( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> ( r3_ideal_1(A,B,C,D)
<=> ( k3_finseq_1(D) = k3_finseq_1(C)
& ! [E] :
( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),C))
=> ( k1_funct_1(C,E) = k1_group_1(A,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)))
& r2_hidden(k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),B) ) ) ) ) ) ) ) ) ).
fof(t39_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m3_ideal_1(C,A,B)
=> ? [D] :
( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
& r3_ideal_1(A,B,C,D) ) ) ) ) ).
fof(t40_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( m3_ideal_1(D,A,C)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
=> ! [G] :
( m2_finseq_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
=> ~ ( r1_tarski(k4_pre_topc(A,B,F,C),E)
& r3_ideal_1(A,C,D,G)
& ! [H] :
( m3_ideal_1(H,B,E)
=> ~ ( k3_finseq_1(D) = k3_finseq_1(H)
& ! [I] :
( r2_hidden(I,k4_relset_1(k5_numbers,u1_struct_0(B),H))
=> k1_funct_1(H,I) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t41_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_ideal_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> m1_ideal_1(k2_partfun1(k5_numbers,u1_struct_0(A),C,k2_finseq_1(D)),A,B) ) ) ) ) ).
fof(t42_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> m2_ideal_1(k2_partfun1(k5_numbers,u1_struct_0(A),C,k2_finseq_1(D)),A,B) ) ) ) ) ).
fof(t43_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m3_ideal_1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> m3_ideal_1(k2_partfun1(k5_numbers,u1_struct_0(A),C,k2_finseq_1(D)),A,B) ) ) ) ) ).
fof(d15_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k7_ideal_1(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,A)
& v2_ideal_1(D,A)
& v3_ideal_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ) ) ).
fof(d16_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k8_ideal_1(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,A)
& v2_ideal_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ) ) ).
fof(d17_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( C = k9_ideal_1(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,A)
& v3_ideal_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ) ) ).
fof(t44_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k7_ideal_1(A,B) = B ) ) ).
fof(t45_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k8_ideal_1(A,B) = B ) ) ).
fof(t46_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k9_ideal_1(A,B) = B ) ) ).
fof(d18_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( m4_ideal_1(C,A,B)
<=> k7_ideal_1(A,C) = B ) ) ) ) ).
fof(t47_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> k7_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) ).
fof(t48_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> k7_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) ).
fof(t49_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> k8_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) ).
fof(t50_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> k9_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) ).
fof(t51_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> k7_ideal_1(A,k1_struct_0(A,k2_group_1(A))) = u1_struct_0(A) ) ).
fof(t52_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& l3_vectsp_1(A) )
=> k8_ideal_1(A,k1_struct_0(A,k2_group_1(A))) = u1_struct_0(A) ) ).
fof(t53_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> k9_ideal_1(A,k1_struct_0(A,k2_group_1(A))) = u1_struct_0(A) ) ).
fof(t54_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> k7_ideal_1(A,k2_pre_topc(A)) = u1_struct_0(A) ) ).
fof(t55_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> k8_ideal_1(A,k2_pre_topc(A)) = u1_struct_0(A) ) ).
fof(t56_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> k9_ideal_1(A,k2_pre_topc(A)) = u1_struct_0(A) ) ).
fof(t57_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,C)
=> r1_tarski(k7_ideal_1(A,B),k7_ideal_1(A,C)) ) ) ) ) ).
fof(t58_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,C)
=> r1_tarski(k8_ideal_1(A,B),k8_ideal_1(A,C)) ) ) ) ) ).
fof(t59_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(B,C)
=> r1_tarski(k9_ideal_1(A,B),k9_ideal_1(A,C)) ) ) ) ) ).
fof(t60_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( r2_hidden(C,k7_ideal_1(A,B))
<=> ? [D] :
( m1_ideal_1(D,A,B)
& C = k9_rlvect_1(A,D) ) ) ) ) ).
fof(t61_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( r2_hidden(C,k8_ideal_1(A,B))
<=> ? [D] :
( m2_ideal_1(D,A,B)
& C = k9_rlvect_1(A,D) ) ) ) ) ).
fof(t62_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( r2_hidden(C,k9_ideal_1(A,B))
<=> ? [D] :
( m3_ideal_1(D,A,B)
& C = k9_rlvect_1(A,D) ) ) ) ) ).
fof(t63_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( k7_ideal_1(A,B) = k8_ideal_1(A,B)
& k7_ideal_1(A,B) = k9_ideal_1(A,B) ) ) ) ).
fof(t66_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(B,k7_ideal_1(A,k1_struct_0(A,B))) ) ) ).
fof(t67_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k7_ideal_1(A,B))
=> r1_tarski(k7_ideal_1(A,k1_struct_0(A,C)),k7_ideal_1(A,B)) ) ) ) ) ).
fof(t68_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(B,k7_ideal_1(A,k2_struct_0(A,B,C)))
& r2_hidden(C,k7_ideal_1(A,k2_struct_0(A,B,C))) ) ) ) ) ).
fof(t69_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_tarski(k7_ideal_1(A,k1_struct_0(A,B)),k7_ideal_1(A,k2_struct_0(A,B,C)))
& r1_tarski(k7_ideal_1(A,k1_struct_0(A,C)),k7_ideal_1(A,k2_struct_0(A,B,C))) ) ) ) ) ).
fof(t70_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k10_ideal_1(A,B,k1_rlvect_1(A)) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ).
fof(t71_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k10_ideal_1(A,B,k2_group_1(A)) = B ) ) ).
fof(t72_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k11_ideal_1(A,B,k11_ideal_1(A,C,D)) = k11_ideal_1(A,k11_ideal_1(A,B,C),D) ) ) ) ) ).
fof(t73_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_tarski(B,k11_ideal_1(A,B,C)) ) ) ) ).
fof(t74_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_tarski(C,k11_ideal_1(A,B,C)) ) ) ) ).
fof(t75_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( v1_ideal_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ( r1_tarski(B,D)
& r1_tarski(C,D) )
=> r1_tarski(k11_ideal_1(A,B,C),D) ) ) ) ) ) ).
fof(t76_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k7_ideal_1(A,k2_struct_0(A,B,C)) = k12_ideal_1(A,k7_ideal_1(A,k1_struct_0(A,B)),k7_ideal_1(A,k1_struct_0(A,C))) ) ) ) ).
fof(t77_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(k13_ideal_1(A,B,C),B)
& r1_tarski(k13_ideal_1(A,B,C),C) ) ) ) ) ).
fof(t78_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k13_ideal_1(A,B,k13_ideal_1(A,C,D)) = k13_ideal_1(A,k13_ideal_1(A,B,C),D) ) ) ) ) ).
fof(t79_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_tarski(D,B)
& r1_tarski(D,C) )
=> r1_tarski(D,k13_ideal_1(A,B,C)) ) ) ) ) ) ).
fof(t80_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& v8_vectsp_1(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r1_tarski(C,B)
=> k13_ideal_1(A,B,k12_ideal_1(A,C,D)) = k12_ideal_1(A,C,k13_ideal_1(A,B,D)) ) ) ) ) ) ).
fof(t81_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> k14_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A)),B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) ).
fof(t82_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_tarski(k14_ideal_1(A,B,C),k13_ideal_1(A,B,C)) ) ) ) ).
fof(t83_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v3_ideal_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> k14_ideal_1(A,B,k12_ideal_1(A,C,D)) = k12_ideal_1(A,k14_ideal_1(A,B,C),k14_ideal_1(A,B,D)) ) ) ) ) ).
fof(t84_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_tarski(k15_ideal_1(A,k12_ideal_1(A,B,C),k13_ideal_1(A,B,C)),k15_ideal_1(A,B,C)) ) ) ) ).
fof(t85_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r1_tarski(k14_ideal_1(A,k11_ideal_1(A,B,C),k13_ideal_1(A,B,C)),k13_ideal_1(A,B,C)) ) ) ) ).
fof(d23_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_ideal_1(A,B,C)
<=> k11_ideal_1(A,B,C) = u1_struct_0(A) ) ) ) ) ).
fof(t86_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r4_ideal_1(A,B,C)
=> r1_tarski(k13_ideal_1(A,B,C),k14_ideal_1(A,k11_ideal_1(A,B,C),k13_ideal_1(A,B,C))) ) ) ) ) ).
fof(t87_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r4_ideal_1(A,B,C)
=> k15_ideal_1(A,B,C) = k13_ideal_1(A,B,C) ) ) ) ) ).
fof(t88_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k16_ideal_1(A,B,C)) ) ) ) ).
fof(t89_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v5_vectsp_1(A)
& v2_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k14_ideal_1(A,k16_ideal_1(A,B,C),C),B) ) ) ) ).
fof(t90_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k14_ideal_1(A,k16_ideal_1(A,B,C),C),B) ) ) ) ).
fof(t91_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k16_ideal_1(A,k16_ideal_1(A,B,C),D) = k16_ideal_1(A,B,k15_ideal_1(A,C,D)) ) ) ) ) ).
fof(t92_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k16_ideal_1(A,k13_ideal_1(A,C,D),B) = k13_ideal_1(A,k16_ideal_1(A,C,B),k16_ideal_1(A,D,B)) ) ) ) ) ).
fof(t93_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v4_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v3_ideal_1(D,A)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
=> k16_ideal_1(A,B,k11_ideal_1(A,C,D)) = k13_ideal_1(A,k16_ideal_1(A,B,C),k16_ideal_1(A,B,D)) ) ) ) ) ).
fof(t94_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k17_ideal_1(A,B))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(k2_binom(A,C,D),k17_ideal_1(A,B)) ) ) ) ) ) ).
fof(t95_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k17_ideal_1(A,k14_ideal_1(A,B,C)) = k17_ideal_1(A,k13_ideal_1(A,B,C)) ) ) ) ).
fof(d26_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v6_ideal_1(B,A)
<=> ? [C] :
( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& B = k7_ideal_1(A,C) ) ) ) ) ).
fof(d27_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v7_ideal_1(A)
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v6_ideal_1(B,A) ) ) ) ).
fof(d28_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v8_ideal_1(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B = k7_ideal_1(A,k1_struct_0(A,C)) ) ) ) ) ).
fof(d29_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v9_ideal_1(A)
<=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> v8_ideal_1(B,A) ) ) ) ).
fof(t96_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( B != k1_struct_0(A,k1_rlvect_1(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( C != k1_rlvect_1(A)
& r2_hidden(C,B) ) ) ) ) ) ).
fof(t97_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_int_3(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> v9_ideal_1(A) ) ).
fof(t98_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v9_ideal_1(A)
=> v7_ideal_1(A) ) ) ).
fof(t99_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& v7_ideal_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(C,B)
& k7_ideal_1(A,C) = k7_ideal_1(A,B) ) ) ) ).
fof(t100_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] :
( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& r1_tarski(C,B)
& k7_ideal_1(A,C) = k7_ideal_1(A,B) ) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)),k7_ideal_1(A,k5_relset_1(k5_numbers,u1_struct_0(A),k2_partfun1(k5_numbers,u1_struct_0(A),B,k1_gr_cy_1(k1_nat_1(C,np__1)))))) ) ) ) ) ).
fof(t101_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)),k7_ideal_1(A,k5_relset_1(k5_numbers,u1_struct_0(A),k2_partfun1(k5_numbers,u1_struct_0(A),B,k1_gr_cy_1(k1_nat_1(C,np__1)))))) ) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C))
& v1_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
& v2_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
& v3_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
& m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k1_zfmisc_1(u1_struct_0(A))) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(D,C)
=> r2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,D)) ) ) ) ) ) ) ) ).
fof(t102_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C))
& v1_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
& v2_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
& v3_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
& m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k1_zfmisc_1(u1_struct_0(A))) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(D,C)
=> r2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,D)) ) ) ) ) )
=> v7_ideal_1(A) ) ) ).
fof(dt_m1_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_ideal_1(C,A,B)
=> m2_finseq_1(C,u1_struct_0(A)) ) ) ).
fof(existence_m1_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m1_ideal_1(C,A,B) ) ).
fof(dt_m2_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m2_ideal_1(C,A,B)
=> m2_finseq_1(C,u1_struct_0(A)) ) ) ).
fof(existence_m2_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m2_ideal_1(C,A,B) ) ).
fof(dt_m3_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m3_ideal_1(C,A,B)
=> m2_finseq_1(C,u1_struct_0(A)) ) ) ).
fof(existence_m3_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m3_ideal_1(C,A,B) ) ).
fof(dt_m4_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m4_ideal_1(C,A,B)
=> ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ).
fof(existence_m4_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m4_ideal_1(C,A,B) ) ).
fof(redefinition_v4_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& v2_ideal_1(B,A)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v4_ideal_1(B,A)
<=> v1_realset1(B) ) ) ).
fof(dt_k1_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k1_ideal_1(A,B))
& v1_funct_2(k1_ideal_1(A,B),k2_zfmisc_1(B,B),B)
& m2_relset_1(k1_ideal_1(A,B),k2_zfmisc_1(B,B),B) ) ) ).
fof(dt_k2_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& ~ v1_xboole_0(B)
& v3_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k2_ideal_1(A,B))
& v1_funct_2(k2_ideal_1(A,B),k2_zfmisc_1(B,B),B)
& m2_relset_1(k2_ideal_1(A,B),k2_zfmisc_1(B,B),B) ) ) ).
fof(dt_k3_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& ~ v1_xboole_0(B)
& v1_ideal_1(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v3_struct_0(k3_ideal_1(A,B))
& l1_rlvect_1(k3_ideal_1(A,B)) ) ) ).
fof(dt_k4_ideal_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_ideal_1(D,A,B)
& m1_ideal_1(E,A,C) )
=> m1_ideal_1(k4_ideal_1(A,B,C,D,E),A,k4_subset_1(u1_struct_0(A),B,C)) ) ).
fof(redefinition_k4_ideal_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m1_ideal_1(D,A,B)
& m1_ideal_1(E,A,C) )
=> k4_ideal_1(A,B,C,D,E) = k7_finseq_1(D,E) ) ).
fof(dt_k5_ideal_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m2_ideal_1(D,A,B)
& m2_ideal_1(E,A,C) )
=> m2_ideal_1(k5_ideal_1(A,B,C,D,E),A,k4_subset_1(u1_struct_0(A),B,C)) ) ).
fof(redefinition_k5_ideal_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m2_ideal_1(D,A,B)
& m2_ideal_1(E,A,C) )
=> k5_ideal_1(A,B,C,D,E) = k7_finseq_1(D,E) ) ).
fof(dt_k6_ideal_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m3_ideal_1(D,A,B)
& m3_ideal_1(E,A,C) )
=> m3_ideal_1(k6_ideal_1(A,B,C,D,E),A,k4_subset_1(u1_struct_0(A),B,C)) ) ).
fof(redefinition_k6_ideal_1,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
& m3_ideal_1(D,A,B)
& m3_ideal_1(E,A,C) )
=> k6_ideal_1(A,B,C,D,E) = k7_finseq_1(D,E) ) ).
fof(dt_k7_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k7_ideal_1(A,B))
& v1_ideal_1(k7_ideal_1(A,B),A)
& v2_ideal_1(k7_ideal_1(A,B),A)
& v3_ideal_1(k7_ideal_1(A,B),A)
& m1_subset_1(k7_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k8_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k8_ideal_1(A,B))
& v1_ideal_1(k8_ideal_1(A,B),A)
& v2_ideal_1(k8_ideal_1(A,B),A)
& m1_subset_1(k8_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k9_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( ~ v1_xboole_0(k9_ideal_1(A,B))
& v1_ideal_1(k9_ideal_1(A,B),A)
& v3_ideal_1(k9_ideal_1(A,B),A)
& m1_subset_1(k9_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k10_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k10_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k11_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k11_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k12_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k12_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(commutativity_k12_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k12_ideal_1(A,B,C) = k12_ideal_1(A,C,B) ) ).
fof(redefinition_k12_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k12_ideal_1(A,B,C) = k11_ideal_1(A,B,C) ) ).
fof(dt_k13_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k13_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(commutativity_k13_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k13_ideal_1(A,B,C) = k13_ideal_1(A,C,B) ) ).
fof(idempotence_k13_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k13_ideal_1(A,B,B) = B ) ).
fof(redefinition_k13_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k13_ideal_1(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k14_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k14_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k15_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k15_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(commutativity_k15_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k15_ideal_1(A,B,C) = k15_ideal_1(A,C,B) ) ).
fof(redefinition_k15_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k15_ideal_1(A,B,C) = k14_ideal_1(A,B,C) ) ).
fof(dt_k16_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k16_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k17_ideal_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k17_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(t64_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k7_ideal_1(A,k1_struct_0(A,B)) = a_2_0_ideal_1(A,B) ) ) ).
fof(t65_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_vectsp_1(A)
& v1_binom(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k7_ideal_1(A,k2_struct_0(A,B,C)) = a_3_0_ideal_1(A,B,C) ) ) ) ).
fof(d19_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k10_ideal_1(A,B,C) = a_3_1_ideal_1(A,B,C) ) ) ) ).
fof(d20_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k11_ideal_1(A,B,C) = a_3_2_ideal_1(A,B,C) ) ) ) ).
fof(d21_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k13_ideal_1(A,B,C) = a_3_3_ideal_1(A,B,C) ) ) ) ).
fof(d22_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k14_ideal_1(A,B,C) = a_3_4_ideal_1(A,B,C) ) ) ) ).
fof(d24_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k16_ideal_1(A,B,C) = a_3_5_ideal_1(A,B,C) ) ) ) ).
fof(d25_ideal_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k17_ideal_1(A,B) = a_2_1_ideal_1(A,B) ) ) ).
fof(fraenkel_a_2_0_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v7_vectsp_1(B)
& v1_binom(B)
& v1_algstr_1(B)
& l3_vectsp_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_ideal_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = k10_group_1(B,C,D) ) ) ) ).
fof(fraenkel_a_3_0_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v7_vectsp_1(B)
& v1_binom(B)
& v1_algstr_1(B)
& l3_vectsp_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_ideal_1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(B))
& m1_subset_1(F,u1_struct_0(B))
& A = k4_rlvect_1(B,k10_group_1(B,C,E),k10_group_1(B,D,F)) ) ) ) ).
fof(fraenkel_a_3_1_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_group_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_1_ideal_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k1_group_1(B,D,E)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_2_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_rlvect_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_2_ideal_1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,u1_struct_0(B))
& m1_subset_1(F,u1_struct_0(B))
& A = k2_rlvect_1(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_3_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_struct_0(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_3_ideal_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r2_hidden(E,C)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_4_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_4_ideal_1(B,C,D))
<=> ? [E] :
( m2_finseq_1(E,u1_struct_0(B))
& A = k9_rlvect_1(B,E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,F)
& r1_xreal_0(F,k3_finseq_1(E))
& ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(B))
=> ~ ( k1_funct_1(E,F) = k1_group_1(B,G,H)
& r2_hidden(G,C)
& r2_hidden(H,D) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_3_5_ideal_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_group_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_3_5_ideal_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = E
& r1_tarski(k10_ideal_1(B,D,E),C) ) ) ) ).
fof(fraenkel_a_2_1_ideal_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& l3_vectsp_1(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_1_ideal_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r2_hidden(k2_binom(B,D,E),C) ) ) ) ) ).
%------------------------------------------------------------------------------