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Metamath Proof Explorer

Theorem cdlemk19xlem

Description: Lemma for cdlemk19x . (Contributed by NM, 30-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b 
|- B = ( Base ` K )
cdlemk5.l 
|- .<_ = ( le ` K )
cdlemk5.j 
|- .\/ = ( join ` K )
cdlemk5.m 
|- ./\ = ( meet ` K )
cdlemk5.a 
|- A = ( Atoms ` K )
cdlemk5.h 
|- H = ( LHyp ` K )
cdlemk5.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r 
|- R = ( ( trL ` K ) ` W )
cdlemk5.z 
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y 
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.x 
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
Assertion cdlemk19xlem 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) )

Proof

Step Hyp Ref Expression
1 cdlemk5.b 
 |-  B = ( Base ` K )
2 cdlemk5.l 
 |-  .<_ = ( le ` K )
3 cdlemk5.j 
 |-  .\/ = ( join ` K )
4 cdlemk5.m 
 |-  ./\ = ( meet ` K )
5 cdlemk5.a 
 |-  A = ( Atoms ` K )
6 cdlemk5.h 
 |-  H = ( LHyp ` K )
7 cdlemk5.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk5.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk5.z 
 |-  Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10 cdlemk5.y 
 |-  Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11 cdlemk5.x 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( K e. HL /\ W e. H ) )
13 simp2l1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> F e. T )
14 simp2l2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> F =/= ( _I |` B ) )
15 13 14 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
16 simp2l3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> N e. T )
17 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
18 simp1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( R ` F ) = ( R ` N ) )
19 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> b e. T )
20 simp3rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> b =/= ( _I |` B ) )
21 simp3rr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
22 20 21 21 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` F ) ) )
23 1 2 3 4 5 6 7 8 9 10 11 cdlemk42 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( F e. T /\ F =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( [_ F / g ]_ X ` P ) = [_ F / g ]_ Y )
24 12 15 15 16 17 18 19 22 23 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( [_ F / g ]_ X ` P ) = [_ F / g ]_ Y )
25 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) )
26 1 2 3 4 5 6 7 8 9 10 cdlemk19y 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> [_ F / g ]_ Y = ( N ` P ) )
27 12 15 16 17 18 25 26 syl231anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> [_ F / g ]_ Y = ( N ` P ) )
28 24 27 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) ) ) ) -> ( [_ F / g ]_ X ` P ) = ( N ` P ) )