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Metamath Proof Explorer

Theorem cdlemkyyN

Description: Part of proof of Lemma K of Crawley p. 118. TODO: clean up ( b Y G ) stuff. (Contributed by NM, 21-Jul-2013) (New usage is discouraged.)

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 ) )
cdlemk5a.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk5a.u1 
|- V = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
Assertion cdlemkyyN 
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = ( ( b V G ) ` 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 cdlemk5a.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
13 cdlemk5a.u1 
 |-  V = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
14 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> K e. HL )
15 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> W e. H )
16 14 15 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
17 simp13 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` F ) = ( R ` N ) )
18 simp211 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F e. T )
19 simp3l 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b e. T )
20 simp213 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> N e. T )
21 simp3r2 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
22 simp212 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> F =/= ( _I |` B ) )
23 simp3r1 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b =/= ( _I |` B ) )
24 22 23 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) )
25 simp23 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
26 1 2 3 4 5 6 7 8 12 cdlemk30 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ b e. T /\ N e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( S ` b ) ` P ) = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) )
27 16 17 18 19 20 21 24 25 26 syl233anc 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( S ` b ) ` P ) = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) )
28 27 9 eqtr4di 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( S ` b ) ` P ) = Z )
29 28 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) = ( Z .\/ ( R ` ( G o. `' b ) ) ) )
30 29 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
31 18 19 20 3jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ b e. T /\ N e. T ) )
32 simp22l 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G e. T )
33 simp3r3 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( R ` b ) =/= ( R ` G ) )
34 21 33 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) )
35 simp22r 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G =/= ( _I |` B ) )
36 22 23 35 3jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F =/= ( _I |` B ) /\ b =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
37 1 2 3 4 5 6 7 8 12 13 cdlemk31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ b e. T /\ N e. T ) /\ G e. T ) /\ ( ( ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ b =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( b V G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
38 16 17 31 32 34 36 25 37 syl223anc 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( b V G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( G o. `' b ) ) ) ) )
39 18 22 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
40 simp22 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( G e. T /\ G =/= ( _I |` B ) ) )
41 simp3 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
42 1 2 3 4 5 6 7 8 9 10 11 cdlemk42yN 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _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 ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
43 16 39 40 20 25 17 41 42 syl331anc 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) )
44 30 38 43 3eqtr4rd 
 |-  ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( [_ G / g ]_ X ` P ) = ( ( b V G ) ` P ) )