TPTP Problem File: ALG215+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ALG215+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : General Algebra
% Problem : Linear Independence in Right Module over Domain T04
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t4_rmod_5 [Urb08]
% Status : Theorem
% Rating : 0.42 v8.2.0, 0.36 v7.5.0, 0.44 v7.4.0, 0.30 v7.3.0, 0.38 v7.2.0, 0.34 v7.1.0, 0.30 v7.0.0, 0.33 v6.4.0, 0.38 v6.2.0, 0.48 v6.1.0, 0.50 v6.0.0, 0.43 v5.5.0, 0.59 v5.4.0, 0.61 v5.3.0, 0.67 v5.2.0, 0.55 v5.1.0, 0.57 v5.0.0, 0.58 v4.1.0, 0.61 v4.0.0, 0.62 v3.7.0, 0.65 v3.5.0, 0.63 v3.4.0
% Syntax : Number of formulae : 77 ( 25 unt; 0 def)
% Number of atoms : 357 ( 11 equ)
% Maximal formula atoms : 22 ( 4 avg)
% Number of connectives : 332 ( 52 ~; 1 |; 215 &)
% ( 7 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-4 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-4 aty)
% Number of variables : 147 ( 122 !; 25 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t4_rmod_5,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> v1_rmod_5(k1_subset_1(u1_struct_0(B)),A,B) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(d1_rmod_5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v1_rmod_5(C,A,B)
<=> ! [D] :
( m2_rmod_4(D,A,B,C)
=> ( k5_rmod_4(A,B,D) = k1_rlvect_1(B)
=> k2_rmod_4(A,B,D) = k1_xboole_0 ) ) ) ) ) ) ).
fof(d5_rmod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_4(C,A,B)
=> k2_rmod_4(A,B,C) = a_3_0_rmod_4(A,B,C) ) ) ) ).
fof(d7_rmod_4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_rmod_4(D,A,B)
=> ( m2_rmod_4(D,A,B,C)
<=> r1_tarski(k2_rmod_4(A,B,D),C) ) ) ) ) ) ).
fof(dt_k1_fraenkel,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> m1_fraenkel(k1_fraenkel(A,B),A,B) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_funct_2,axiom,
$true ).
fof(dt_k1_rlvect_1,axiom,
! [A] :
( l2_struct_0(A)
=> m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ).
fof(dt_k1_subset_1,axiom,
! [A] :
( v1_xboole_0(k1_subset_1(A))
& m1_subset_1(k1_subset_1(A),k1_zfmisc_1(A)) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_rmod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_rmod_4(C,A,B) )
=> ( v1_finset_1(k2_rmod_4(A,B,C))
& m1_subset_1(k2_rmod_4(A,B,C),k1_zfmisc_1(u1_struct_0(B))) ) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k5_rmod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_rmod_4(C,A,B) )
=> m1_subset_1(k5_rmod_4(A,B,C),u1_struct_0(B)) ) ).
fof(dt_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> m1_subset_1(k8_funct_2(A,B,C,D),B) ) ).
fof(dt_l1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_rlvect_1,axiom,
! [A] :
( l1_rlvect_1(A)
=> l2_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_l1_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> l1_group_1(A) ) ).
fof(dt_l1_vectsp_2,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_vectsp_2(B,A)
=> l1_rlvect_1(B) ) ) ).
fof(dt_l2_struct_0,axiom,
! [A] :
( l2_struct_0(A)
=> l1_struct_0(A) ) ).
fof(dt_l2_vectsp_1,axiom,
! [A] :
( l2_vectsp_1(A)
=> ( l1_vectsp_1(A)
& l2_struct_0(A) ) ) ).
fof(dt_l3_vectsp_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( l1_rlvect_1(A)
& l2_vectsp_1(A) ) ) ).
fof(dt_m1_fraenkel,axiom,
! [A,B,C] :
( m1_fraenkel(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_rmod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ! [C] :
( m1_rmod_4(C,A,B)
=> m2_fraenkel(C,u1_struct_0(B),u1_struct_0(A),k1_fraenkel(u1_struct_0(B),u1_struct_0(A))) ) ) ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ! [D] :
( m2_fraenkel(D,A,B,C)
=> ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) ) ) ) ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_m2_rmod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [D] :
( m2_rmod_4(D,A,B,C)
=> m1_rmod_4(D,A,B) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_group_1,axiom,
? [A] : l1_group_1(A) ).
fof(existence_l1_rlvect_1,axiom,
? [A] : l1_rlvect_1(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_l1_vectsp_1,axiom,
? [A] : l1_vectsp_1(A) ).
fof(existence_l1_vectsp_2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] : l1_vectsp_2(B,A) ) ).
fof(existence_l2_struct_0,axiom,
? [A] : l2_struct_0(A) ).
fof(existence_l2_vectsp_1,axiom,
? [A] : l2_vectsp_1(A) ).
fof(existence_l3_vectsp_1,axiom,
? [A] : l3_vectsp_1(A) ).
fof(existence_m1_fraenkel,axiom,
! [A,B] :
? [C] : m1_fraenkel(C,A,B) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_rmod_4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A) )
=> ? [C] : m1_rmod_4(C,A,B) ) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ? [D] : m2_fraenkel(D,A,B,C) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(existence_m2_rmod_4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_2(B,A)
& l1_vectsp_2(B,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ? [D] : m2_rmod_4(D,A,B,C) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(fraenkel_a_3_0_rmod_4,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B)
& ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v5_vectsp_2(C,B)
& l1_vectsp_2(C,B)
& m1_rmod_4(D,B,C) )
=> ( r2_hidden(A,a_3_0_rmod_4(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(C))
& A = E
& k8_funct_2(u1_struct_0(C),u1_struct_0(B),D,E) != k1_rlvect_1(B) ) ) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_rlvect_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc4_struct_0,axiom,
? [A] :
( l2_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(rc9_vectsp_2,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_vectsp_2(B,A)
& ~ v3_struct_0(B) ) ) ).
fof(redefinition_k1_fraenkel,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> k1_fraenkel(A,B) = k1_funct_2(A,B) ) ).
fof(redefinition_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(redefinition_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ! [D] :
( m2_fraenkel(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t2_tarski,axiom,
! [A,B] :
( ! [C] :
( r2_hidden(C,A)
<=> r2_hidden(C,B) )
=> A = B ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t3_xboole_1,axiom,
! [A] :
( r1_tarski(A,k1_xboole_0)
=> A = k1_xboole_0 ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
%------------------------------------------------------------------------------