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Metamath Proof Explorer

Theorem cdlemk52

Description: Part of proof of Lemma K of Crawley p. 118. Line 6, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 23-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 cdlemk52 
|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) = ( [_ ( G o. I ) / g ]_ X ` 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 simp11l 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> K e. HL )
13 12 hllatd 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> K e. Lat )
14 simp11 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp12 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
16 simp13 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G e. T /\ G =/= ( _I |` B ) ) )
17 simp21 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> N e. T )
18 simp22 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( P e. A /\ -. P .<_ W ) )
19 simp23 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` F ) = ( R ` N ) )
20 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s 
 |-  ( ( ( 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 ) ) ) -> [_ G / g ]_ X e. T )
21 14 15 16 17 18 19 20 syl132anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ G / g ]_ X e. T )
22 simp31 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> I e. T )
23 simp32 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> I =/= ( _I |` B ) )
24 22 23 jca 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) ) )
25 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ I / g ]_ X e. T )
26 14 15 24 17 18 19 25 syl132anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ I / g ]_ X e. T )
27 6 7 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T /\ [_ I / g ]_ X e. T ) -> ( [_ G / g ]_ X o. [_ I / g ]_ X ) e. T )
28 14 21 26 27 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X o. [_ I / g ]_ X ) e. T )
29 simp22l 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> P e. A )
30 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( [_ G / g ]_ X o. [_ I / g ]_ X ) e. T /\ P e. A ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A )
31 14 28 29 30 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A )
32 1 5 atbase 
 |-  ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. B )
33 31 32 syl 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. B )
34 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T /\ P e. A ) -> ( [_ G / g ]_ X ` P ) e. A )
35 14 21 29 34 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) e. A )
36 1 5 atbase 
 |-  ( ( [_ G / g ]_ X ` P ) e. A -> ( [_ G / g ]_ X ` P ) e. B )
37 35 36 syl 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) e. B )
38 1 6 7 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ I / g ]_ X e. T ) -> ( R ` [_ I / g ]_ X ) e. B )
39 14 26 38 syl2anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` [_ I / g ]_ X ) e. B )
40 1 3 latjcl 
 |-  ( ( K e. Lat /\ ( [_ G / g ]_ X ` P ) e. B /\ ( R ` [_ I / g ]_ X ) e. B ) -> ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) e. B )
41 13 37 39 40 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) e. B )
42 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ I / g ]_ X e. T /\ P e. A ) -> ( [_ I / g ]_ X ` P ) e. A )
43 14 26 29 42 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ I / g ]_ X ` P ) e. A )
44 1 5 atbase 
 |-  ( ( [_ I / g ]_ X ` P ) e. A -> ( [_ I / g ]_ X ` P ) e. B )
45 43 44 syl 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ I / g ]_ X ` P ) e. B )
46 1 6 7 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T ) -> ( R ` [_ G / g ]_ X ) e. B )
47 14 21 46 syl2anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` [_ G / g ]_ X ) e. B )
48 1 3 latjcl 
 |-  ( ( K e. Lat /\ ( [_ I / g ]_ X ` P ) e. B /\ ( R ` [_ G / g ]_ X ) e. B ) -> ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) e. B )
49 13 45 47 48 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) e. B )
50 1 4 latmcl 
 |-  ( ( K e. Lat /\ ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) e. B /\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) e. B ) -> ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) ) e. B )
51 13 41 49 50 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) ) e. B )
52 simp11r 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> W e. H )
53 1 5 6 7 8 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ I e. T /\ I =/= ( _I |` B ) ) -> ( R ` I ) e. A )
54 12 52 22 23 53 syl211anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` I ) e. A )
55 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ ( [_ G / g ]_ X ` P ) e. A /\ ( R ` I ) e. A ) -> ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) e. B )
56 12 35 54 55 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) e. B )
57 simp13l 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> G e. T )
58 simp13r 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> G =/= ( _I |` B ) )
59 1 5 6 7 8 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
60 12 52 57 58 59 syl211anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` G ) e. A )
61 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ ( [_ I / g ]_ X ` P ) e. A /\ ( R ` G ) e. A ) -> ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) e. B )
62 12 43 60 61 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) e. B )
63 1 4 latmcl 
 |-  ( ( K e. Lat /\ ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) e. B /\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) e. B ) -> ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) e. B )
64 13 56 62 63 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) e. B )
65 1 2 3 4 5 6 7 8 9 10 11 cdlemk50 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) ) )
66 24 65 syld3an3 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) ) )
67 1 2 3 4 5 6 7 8 9 10 11 cdlemk51 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) ) .<_ ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )
68 24 67 syld3an3 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` [_ I / g ]_ X ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) ) .<_ ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )
69 1 2 13 33 51 64 66 68 lattrd 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )
70 1 2 3 4 5 6 7 8 9 10 11 cdlemk47 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) = ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )
71 69 70 breqtrrd 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( [_ ( G o. I ) / g ]_ X ` P ) )
72 hlatl 
 |-  ( K e. HL -> K e. AtLat )
73 12 72 syl 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> K e. AtLat )
74 6 7 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ I e. T ) -> ( G o. I ) e. T )
75 14 57 22 74 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G o. I ) e. T )
76 57 22 jca 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G e. T /\ I e. T ) )
77 simp33 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` G ) =/= ( R ` I ) )
78 1 6 7 8 trlconid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ I e. T ) /\ ( R ` G ) =/= ( R ` I ) ) -> ( G o. I ) =/= ( _I |` B ) )
79 14 76 77 78 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G o. I ) =/= ( _I |` B ) )
80 75 79 jca 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( G o. I ) e. T /\ ( G o. I ) =/= ( _I |` B ) ) )
81 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( ( G o. I ) e. T /\ ( G o. I ) =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ ( G o. I ) / g ]_ X e. T )
82 14 15 80 17 18 19 81 syl132anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X e. T )
83 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ ( G o. I ) / g ]_ X e. T /\ P e. A ) -> ( [_ ( G o. I ) / g ]_ X ` P ) e. A )
84 14 82 29 83 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) e. A )
85 2 5 atcmp 
 |-  ( ( K e. AtLat /\ ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A /\ ( [_ ( G o. I ) / g ]_ X ` P ) e. A ) -> ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( [_ ( G o. I ) / g ]_ X ` P ) <-> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) = ( [_ ( G o. I ) / g ]_ X ` P ) ) )
86 73 31 84 85 syl3anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( [_ ( G o. I ) / g ]_ X ` P ) <-> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) = ( [_ ( G o. I ) / g ]_ X ` P ) ) )
87 71 86 mpbid 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) = ( [_ ( G o. I ) / g ]_ X ` P ) )