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Theorem cdlemk25-3 36192
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the ( R ` C ) = ( R ` D ) requirement from cdlemk24-3 36191. (Contributed by NM, 7-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b |- B = ( Base ` K )
cdlemk3.l |- .<_ = ( le ` K )
cdlemk3.j |- .\/ = ( join ` K )
cdlemk3.m |- ./\ = ( meet ` K )
cdlemk3.a |- A = ( Atoms ` K )
cdlemk3.h |- H = ( LHyp ` K )
cdlemk3.t |- T = ( ( LTrn ` K ) ` W )
cdlemk3.r |- R = ( ( trL ` K ) ` W )
cdlemk3.s |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk3.u1 |- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
Assertion
Ref Expression
cdlemk25-3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
Distinct variable groups:   e, d, f, i, ./\   .<_ , i   .\/ , d, e, f, i   A, i   j, d, D, e, f, i   f, F, i   G, d, e, j   i, H   i, K   f, N, i   P, d, e, f, i   R, d, e, f, i   T, d, e, f, i   W, d, e, f, i   ./\ , j   .<_ , j   .\/ , j   A, j   j, F   j, H   j, K   j, N   P, j   R, j   S, d, e, j   T, j   j, W   F, d, e   .<_ , e   C, d, e, f, i, j   f, G, i   x, d, e, f, i, j
Allowed substitution hints:   A( x, e, f, d)   B( x, e, f, i, j, d)   C( x)   D( x)   P( x)   R( x)   S( x, f, i)   T( x)   F( x)   G( x)   H( x, e, f, d)   .\/ ( x)   K( x, e, f, d)   .<_ ( x, f, d)   ./\ ( x)   N( x, e, d)   W( x)   Y( x, e, f, i, j, d)
Proof of Theorem cdlemk25-3
StepHypRef Expression
1 simpl1 1064 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) )
2 simpl2 1065 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) )
3 simpl31 1142 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
4 simpl32 1143 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( R ` G ) =/= ( R ` D ) )
5 simpr 477 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( R ` C ) = ( R ` D ) )
64, 5jca 554 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) )
7 simpl33 1144 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) )
8 cdlemk3.b . . . 4 |- B = ( Base ` K )
9 cdlemk3.l . . . 4 |- .<_ = ( le ` K )
10 cdlemk3.j . . . 4 |- .\/ = ( join ` K )
11 cdlemk3.m . . . 4 |- ./\ = ( meet ` K )
12 cdlemk3.a . . . 4 |- A = ( Atoms ` K )
13 cdlemk3.h . . . 4 |- H = ( LHyp ` K )
14 cdlemk3.t . . . 4 |- T = ( ( LTrn ` K ) ` W )
15 cdlemk3.r . . . 4 |- R = ( ( trL ` K ) ` W )
16 cdlemk3.s . . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
17 cdlemk3.u1 . . . 4 |- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
188, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk24-3 36191 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
191, 2, 3, 6, 7, 18syl113anc 1338 . 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
20 simp11 1091 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( K e. HL /\ W e. H ) )
21 simp121 1193 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> F e. T )
22 simp122 1194 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> D e. T )
2320, 21, 223jca 1242 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
2423adantr 481 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
25 simp123 1195 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> N e. T )
26 simp131 1196 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> G e. T )
27 simp132 1197 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> C e. T )
2825, 26, 273jca 1242 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( N e. T /\ G e. T /\ C e. T ) )
29 simp21 1094 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
30 simp221 1202 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` F ) = ( R ` N ) )
3128, 29, 303jca 1242 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( N e. T /\ G e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
3231adantr 481 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( N e. T /\ G e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
33 simp222 1203 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> F =/= ( _I |` B ) )
34 simp223 1204 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> D =/= ( _I |` B ) )
35 simp231 1205 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> G =/= ( _I |` B ) )
3633, 34, 353jca 1242 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
3736adantr 481 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
38 simp232 1206 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> C =/= ( _I |` B ) )
39 simp311 1208 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` G ) =/= ( R ` C ) )
40 simp312 1209 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` C ) =/= ( R ` F ) )
4138, 39, 403jca 1242 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) ) )
4241adantr 481 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) ) )
43 simp313 1210 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
4443adantr 481 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( R ` D ) =/= ( R ` F ) )
45 simpl32 1143 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( R ` G ) =/= ( R ` D ) )
46 simpr 477 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( R ` C ) =/= ( R ` D ) )
4744, 45, 463jca 1242 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) )
488, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemk22-3 36189 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
4924, 32, 37, 42, 47, 48syl113anc 1338 . 2 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
5019, 49pm2.61dane 2881 1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446
This theorem is referenced by:  cdlemk26-3  36194
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