SET007 Axioms: SET007+243.ax
%------------------------------------------------------------------------------
% File : SET007+243 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : From Loops to Abelian Multiplicative Groups with Zero
% 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 : algstr_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 98 ( 33 unt; 0 def)
% Number of atoms : 656 ( 109 equ)
% Maximal formula atoms : 29 ( 6 avg)
% Number of connectives : 658 ( 100 ~; 4 |; 332 &)
% ( 21 <=>; 201 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 1 prp; 0-2 aty)
% Number of functors : 24 ( 24 usr; 9 con; 0-4 aty)
% Number of variables : 212 ( 189 !; 23 ?)
% 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(fc1_algstr_1,axiom,
( ~ v3_struct_0(k2_algstr_1)
& v1_rlvect_1(k2_algstr_1) ) ).
fof(cc1_algstr_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A) ) ) ) ).
fof(cc2_algstr_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v6_algstr_1(A) ) ) ) ).
fof(fc2_algstr_1,axiom,
( ~ v3_struct_0(k2_algstr_1)
& v1_rlvect_1(k2_algstr_1)
& v4_rlvect_1(k2_algstr_1)
& v5_rlvect_1(k2_algstr_1)
& v1_algstr_1(k2_algstr_1)
& v2_algstr_1(k2_algstr_1)
& v3_algstr_1(k2_algstr_1)
& v4_algstr_1(k2_algstr_1)
& v5_algstr_1(k2_algstr_1)
& v6_algstr_1(k2_algstr_1) ) ).
fof(rc1_algstr_1,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v1_rlvect_1(A)
& v5_rlvect_1(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) ) ).
fof(rc2_algstr_1,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v1_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(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) ) ).
fof(cc3_algstr_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v6_rlvect_1(A) ) ) ) ).
fof(cc4_algstr_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A) )
=> ( ~ v3_struct_0(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) ) ) ) ).
fof(fc3_algstr_1,axiom,
( ~ v3_struct_0(k2_algstr_1)
& v1_rlvect_1(k2_algstr_1)
& v3_rlvect_1(k2_algstr_1)
& v4_rlvect_1(k2_algstr_1)
& v5_rlvect_1(k2_algstr_1)
& v6_rlvect_1(k2_algstr_1)
& v1_algstr_1(k2_algstr_1)
& v2_algstr_1(k2_algstr_1)
& v3_algstr_1(k2_algstr_1)
& v4_algstr_1(k2_algstr_1)
& v5_algstr_1(k2_algstr_1)
& v6_algstr_1(k2_algstr_1) ) ).
fof(rc3_algstr_1,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v1_rlvect_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(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) ) ).
fof(fc4_algstr_1,axiom,
( ~ v3_struct_0(k3_algstr_1)
& v1_vectsp_1(k3_algstr_1) ) ).
fof(rc4_algstr_1,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v1_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v7_algstr_1(A)
& v8_algstr_1(A) ) ).
fof(fc5_algstr_1,axiom,
( ~ v3_struct_0(k3_algstr_1)
& v2_group_1(k3_algstr_1)
& v1_vectsp_1(k3_algstr_1)
& v6_vectsp_1(k3_algstr_1)
& v8_vectsp_1(k3_algstr_1)
& v7_algstr_1(k3_algstr_1)
& v8_algstr_1(k3_algstr_1) ) ).
fof(rc5_algstr_1,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v1_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v7_algstr_1(A)
& v8_algstr_1(A) ) ).
fof(fc6_algstr_1,axiom,
( ~ v3_struct_0(k3_algstr_1)
& v2_group_1(k3_algstr_1)
& v4_group_1(k3_algstr_1)
& v1_vectsp_1(k3_algstr_1)
& v6_vectsp_1(k3_algstr_1)
& v8_vectsp_1(k3_algstr_1)
& v7_algstr_1(k3_algstr_1)
& v8_algstr_1(k3_algstr_1) ) ).
fof(rc6_algstr_1,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v1_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v7_algstr_1(A)
& v8_algstr_1(A) ) ).
fof(fc7_algstr_1,axiom,
( ~ v3_struct_0(k6_algstr_1)
& v2_vectsp_1(k6_algstr_1) ) ).
fof(fc8_algstr_1,axiom,
( ~ v3_struct_0(k6_algstr_1)
& v2_group_1(k6_algstr_1)
& v2_vectsp_1(k6_algstr_1)
& v6_vectsp_1(k6_algstr_1)
& v8_vectsp_1(k6_algstr_1) ) ).
fof(cc5_algstr_1,axiom,
! [A] :
( l2_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v11_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v9_algstr_1(A)
& v10_algstr_1(A) ) ) ) ).
fof(rc7_algstr_1,axiom,
? [A] :
( l2_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v2_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& v11_algstr_1(A) ) ).
fof(fc9_algstr_1,axiom,
( ~ v3_struct_0(k6_algstr_1)
& v2_group_1(k6_algstr_1)
& v2_vectsp_1(k6_algstr_1)
& v6_vectsp_1(k6_algstr_1)
& v8_vectsp_1(k6_algstr_1)
& v9_algstr_1(k6_algstr_1)
& v10_algstr_1(k6_algstr_1)
& v11_algstr_1(k6_algstr_1) ) ).
fof(rc8_algstr_1,axiom,
? [A] :
( l2_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v2_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& v11_algstr_1(A) ) ).
fof(fc10_algstr_1,axiom,
( ~ v3_struct_0(k6_algstr_1)
& v2_group_1(k6_algstr_1)
& v4_group_1(k6_algstr_1)
& v2_vectsp_1(k6_algstr_1)
& v6_vectsp_1(k6_algstr_1)
& v8_vectsp_1(k6_algstr_1)
& v9_algstr_1(k6_algstr_1)
& v10_algstr_1(k6_algstr_1)
& v11_algstr_1(k6_algstr_1) ) ).
fof(rc9_algstr_1,axiom,
? [A] :
( l2_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v2_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& v11_algstr_1(A) ) ).
fof(t1_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_rlvect_1(A,D,k1_rlvect_1(A)) = D )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k2_rlvect_1(A,D,E) = k1_rlvect_1(A) ) )
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> k2_rlvect_1(A,k2_rlvect_1(A,D,E),F) = k2_rlvect_1(A,D,k2_rlvect_1(A,E,F)) ) ) )
& k2_rlvect_1(A,B,C) = k1_rlvect_1(A) )
=> k2_rlvect_1(A,C,B) = k1_rlvect_1(A) ) ) ) ) ).
fof(t2_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ( ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,C,k1_rlvect_1(A)) = C )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k2_rlvect_1(A,C,D) = k1_rlvect_1(A) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k2_rlvect_1(A,k2_rlvect_1(A,C,D),E) = k2_rlvect_1(A,C,k2_rlvect_1(A,D,E)) ) ) ) )
=> k2_rlvect_1(A,k1_rlvect_1(A),B) = k2_rlvect_1(A,B,k1_rlvect_1(A)) ) ) ) ).
fof(t3_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) ) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,C,B) = k1_rlvect_1(A) ) ) ) ) ).
fof(d1_algstr_1,axiom,
$true ).
fof(d2_algstr_1,axiom,
$true ).
fof(d3_algstr_1,axiom,
! [A] : k1_algstr_1(A) = A ).
fof(d4_algstr_1,axiom,
k2_algstr_1 = g1_rlvect_1(k1_tarski(k1_xboole_0),k2_midsp_1,k1_algstr_1(k1_xboole_0)) ).
fof(t4_algstr_1,axiom,
$true ).
fof(t5_algstr_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k2_algstr_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_algstr_1))
=> A = B ) ) ).
fof(t6_algstr_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k2_algstr_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k2_algstr_1))
=> k2_rlvect_1(k2_algstr_1,A,B) = k1_rlvect_1(k2_algstr_1) ) ) ).
fof(d5_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v1_algstr_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,k1_rlvect_1(A),B) = B ) ) ) ).
fof(d6_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v2_algstr_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,B,D)
=> C = D ) ) ) ) ) ) ).
fof(d7_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v3_algstr_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k2_rlvect_1(A,C,B) = k2_rlvect_1(A,D,B)
=> C = D ) ) ) ) ) ) ).
fof(d8_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v4_algstr_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k2_rlvect_1(A,D,B) = C ) ) ) ) ) ).
fof(d9_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v5_algstr_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k2_rlvect_1(A,B,D) = C ) ) ) ) ) ).
fof(d10_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v6_algstr_1(A)
<=> ( v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A) ) ) ) ).
fof(t7_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v6_algstr_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k2_rlvect_1(A,B,D) = C ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k2_rlvect_1(A,D,B) = C ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,B,D)
=> C = D ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k2_rlvect_1(A,C,B) = k2_rlvect_1(A,D,B)
=> C = D ) ) ) ) ) ) ) ).
fof(t8_algstr_1,axiom,
$true ).
fof(t9_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& l1_rlvect_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) ) ) ) ) ).
fof(t10_algstr_1,axiom,
$true ).
fof(t11_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v1_algstr_1(A)
& v6_algstr_1(A)
& l1_rlvect_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_rlvect_1(A,B,k1_rlvect_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k2_rlvect_1(A,B,C) = k1_rlvect_1(A) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B) ) ) ) ) ) ).
fof(d11_algstr_1,axiom,
k3_algstr_1 = g1_vectsp_1(k1_tarski(k1_xboole_0),k2_midsp_1,k1_algstr_1(k1_xboole_0)) ).
fof(t12_algstr_1,axiom,
$true ).
fof(t13_algstr_1,axiom,
$true ).
fof(t14_algstr_1,axiom,
$true ).
fof(t15_algstr_1,axiom,
$true ).
fof(t16_algstr_1,axiom,
$true ).
fof(t17_algstr_1,axiom,
$true ).
fof(t18_algstr_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_algstr_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_algstr_1))
=> A = B ) ) ).
fof(t19_algstr_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_algstr_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_algstr_1))
=> k1_group_1(k3_algstr_1,A,B) = k2_group_1(k3_algstr_1) ) ) ).
fof(d12_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v7_algstr_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,B,D) = C ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& k1_group_1(A,D,B) = C ) ) ) ) ) ) ).
fof(d13_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v8_algstr_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
=> C = D ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
=> C = D ) ) ) ) ) ) ) ).
fof(t20_algstr_1,axiom,
$true ).
fof(t21_algstr_1,axiom,
$true ).
fof(t22_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k1_group_1(A,B,C) = k2_group_1(A) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ) ) ).
fof(t23_algstr_1,axiom,
$true ).
fof(t24_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A) )
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& k1_group_1(A,B,C) = k2_group_1(A) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ) ).
fof(d14_algstr_1,axiom,
$true ).
fof(d15_algstr_1,axiom,
$true ).
fof(d16_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k4_algstr_1(A,B)
<=> k1_group_1(A,B,C) = k2_group_1(A) ) ) ) ) ).
fof(t25_algstr_1,axiom,
$true ).
fof(t26_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_group_1(A,B,k4_algstr_1(A,B)) = k2_group_1(A)
& k1_group_1(A,k4_algstr_1(A,B),B) = k2_group_1(A) ) ) ) ).
fof(d17_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_algstr_1(A,B,C) = k1_group_1(A,B,k4_algstr_1(A,C)) ) ) ) ).
fof(d18_algstr_1,axiom,
$true ).
fof(d19_algstr_1,axiom,
$true ).
fof(d20_algstr_1,axiom,
$true ).
fof(d21_algstr_1,axiom,
k6_algstr_1 = g2_vectsp_1(k1_numbers,k35_binop_2,np__1,np__0) ).
fof(t27_algstr_1,axiom,
$true ).
fof(t28_algstr_1,axiom,
$true ).
fof(t29_algstr_1,axiom,
$true ).
fof(t30_algstr_1,axiom,
$true ).
fof(t31_algstr_1,axiom,
$true ).
fof(t32_algstr_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( A != np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> B != k11_binop_2(A,C) ) ) ) ) ).
fof(t33_algstr_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( A != np__0
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> B != k11_binop_2(C,A) ) ) ) ) ).
fof(d22_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v9_algstr_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,B,D) != C ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,D,B) != C ) ) ) ) ) ) ) ).
fof(d23_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v10_algstr_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) ) ) ) ) ).
fof(d24_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v11_algstr_1(A)
<=> ( v9_algstr_1(A)
& v10_algstr_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ) ) ).
fof(t34_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v11_algstr_1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,B,D) != C ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,D,B) != C ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,B,C) = k1_group_1(A,B,D)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_group_1(A,C,B) = k1_group_1(A,D,B)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ) ) ).
fof(t35_algstr_1,axiom,
$true ).
fof(t36_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& ~ v10_vectsp_1(A)
& v11_algstr_1(A)
& l2_vectsp_1(A) )
<=> ( k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) != k2_group_1(A) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ) ) ).
fof(t37_algstr_1,axiom,
$true ).
fof(t38_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& v11_algstr_1(A)
& l2_vectsp_1(A) )
<=> ( k1_rlvect_1(A) != k2_group_1(A)
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) != k2_group_1(A) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = k1_group_1(A,C,B) ) ) ) ) ) ).
fof(d25_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k7_algstr_1(A,B)
<=> k1_group_1(A,B,C) = k2_group_1(A) ) ) ) ) ) ).
fof(t39_algstr_1,axiom,
$true ).
fof(t40_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> ( k1_group_1(A,B,k7_algstr_1(A,B)) = k2_group_1(A)
& k1_group_1(A,k7_algstr_1(A,B),B) = k2_group_1(A) ) ) ) ) ).
fof(d26_algstr_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_algstr_1(A,B,C) = k1_group_1(A,B,k7_algstr_1(A,C)) ) ) ) ).
fof(dt_k1_algstr_1,axiom,
! [A] : m1_subset_1(k1_algstr_1(A),k1_tarski(A)) ).
fof(dt_k2_algstr_1,axiom,
( v1_rlvect_1(k2_algstr_1)
& l1_rlvect_1(k2_algstr_1) ) ).
fof(dt_k3_algstr_1,axiom,
( v1_vectsp_1(k3_algstr_1)
& l1_vectsp_1(k3_algstr_1) ) ).
fof(dt_k4_algstr_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k4_algstr_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k5_algstr_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_algstr_1(A)
& v8_algstr_1(A)
& l1_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k5_algstr_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k6_algstr_1,axiom,
( v2_vectsp_1(k6_algstr_1)
& l2_vectsp_1(k6_algstr_1) ) ).
fof(dt_k7_algstr_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k7_algstr_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k8_algstr_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v9_algstr_1(A)
& v10_algstr_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k8_algstr_1(A,B,C),u1_struct_0(A)) ) ).
%------------------------------------------------------------------------------