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Theorem cdlemk23-3 34513
Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the ( R ` C ) =/= ( R ` D ) requirement from cdlemk22-3 34512. (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
cdlemk23-3 |- ( ( ( ( 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 ` C ) ) /\ ( ( 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 cdlemk23-3
StepHypRef Expression
1 simp11 1044 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( K e. HL /\ W e. H ) )
2 simp121 1146 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> F e. T )
3 simp122 1147 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> D e. T )
4 simp123 1148 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> N e. T )
5 simp131 1149 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> G e. T )
6 simp133 1151 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> x e. T )
74, 5, 63jca 1194 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( N e. T /\ G e. T /\ x e. T ) )
8 simp21 1047 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
9 simp221 1155 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` F ) = ( R ` N ) )
10 simp222 1156 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> F =/= ( _I |` B ) )
11 simp223 1157 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> D =/= ( _I |` B ) )
12 simp231 1158 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> G =/= ( _I |` B ) )
1310, 11, 123jca 1194 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
14 simp233 1160 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> x =/= ( _I |` B ) )
15 simp333 1169 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` G ) =/= ( R ` x ) )
16 simp332 1168 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` x ) =/= ( R ` F ) )
1714, 15, 163jca 1194 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( x =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` x ) /\ ( R ` x ) =/= ( R ` F ) ) )
18 simp313 1163 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
19 simp32l 1139 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
20 simp331 1167 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` x ) =/= ( R ` D ) )
2118, 19, 203jca 1194 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` D ) ) )
22 cdlemk3.b . . . 4 |- B = ( Base ` K )
23 cdlemk3.l . . . 4 |- .<_ = ( le ` K )
24 cdlemk3.j . . . 4 |- .\/ = ( join ` K )
25 cdlemk3.m . . . 4 |- ./\ = ( meet ` K )
26 cdlemk3.a . . . 4 |- A = ( Atoms ` K )
27 cdlemk3.h . . . 4 |- H = ( LHyp ` K )
28 cdlemk3.t . . . 4 |- T = ( ( LTrn ` K ) ` W )
29 cdlemk3.r . . . 4 |- R = ( ( trL ` K ) ` W )
30 cdlemk3.s . . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
31 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 ) ) ) ) ) )
3222, 23, 24, 25, 26, 27, 28, 29, 30, 31cdlemk22-3 34512 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ x e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( x =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` x ) /\ ( R ` x ) =/= ( R ` F ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` D ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( x Y G ) ` P ) )
331, 2, 3, 7, 8, 9, 13, 17, 21, 32syl333anc 1308 . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( x Y G ) ` P ) )
34 simp132 1150 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> C e. T )
35 simp232 1159 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> C =/= ( _I |` B ) )
3610, 35, 123jca 1194 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
37 simp312 1162 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` C ) =/= ( R ` F ) )
38 simp311 1161 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` G ) =/= ( R ` C ) )
39 simp32r 1140 . . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` x ) =/= ( R ` C ) )
4037, 38, 393jca 1194 . . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` x ) =/= ( R ` C ) ) )
4122, 23, 24, 25, 26, 27, 28, 29, 30, 31cdlemk22-3 34512 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( ( N e. T /\ G e. T /\ x e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( x =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` x ) /\ ( R ` x ) =/= ( R ` F ) ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` x ) =/= ( R ` C ) ) ) ) -> ( ( C Y G ) ` P ) = ( ( x Y G ) ` P ) )
421, 2, 34, 7, 8, 9, 36, 17, 40, 41syl333anc 1308 . 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 ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( C Y G ) ` P ) = ( ( x Y G ) ` P ) )
4333, 42eqtr4d 2498 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 ` C ) ) /\ ( ( 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 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   class class class wbr 4415   |-> cmpt 4474   _I cid 4762   `'ccnv 4851   |` cres 4854   o. ccom 4856   ` cfv 5600   iota_crio 6275  (class class class)co 6314   |-> cmpt2 6316   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LHypclh 33593   LTrncltrn 33710   trLctrl 33768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-riotaBAD 32569
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-undef 7045  df-map 7499  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-p1 16334  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108  df-lvols 33109  df-lines 33110  df-psubsp 33112  df-pmap 33113  df-padd 33405  df-lhyp 33597  df-laut 33598  df-ldil 33713  df-ltrn 33714  df-trl 33769
This theorem is referenced by:  cdlemk24-3  34514
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