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Metamath Proof Explorer

Theorem cdlemk12u

Description: Part of proof of Lemma K of Crawley p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma_1 ( U ) case. (Contributed by NM, 4-Jul-2013)

Ref Expression
Hypotheses cdlemk1.b 
|- B = ( Base ` K )
cdlemk1.l 
|- .<_ = ( le ` K )
cdlemk1.j 
|- .\/ = ( join ` K )
cdlemk1.m 
|- ./\ = ( meet ` K )
cdlemk1.a 
|- A = ( Atoms ` K )
cdlemk1.h 
|- H = ( LHyp ` K )
cdlemk1.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r 
|- R = ( ( trL ` K ) ` W )
cdlemk1.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o 
|- O = ( S ` D )
cdlemk1.u 
|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
Assertion cdlemk12u 
|- ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk1.b 
 |-  B = ( Base ` K )
2 cdlemk1.l 
 |-  .<_ = ( le ` K )
3 cdlemk1.j 
 |-  .\/ = ( join ` K )
4 cdlemk1.m 
 |-  ./\ = ( meet ` K )
5 cdlemk1.a 
 |-  A = ( Atoms ` K )
6 cdlemk1.h 
 |-  H = ( LHyp ` K )
7 cdlemk1.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk1.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk1.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk1.o 
 |-  O = ( S ` D )
11 cdlemk1.u 
 |-  U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
12 simp11l 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> K e. HL )
13 simp22l 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> P e. A )
14 simp11 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp212 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> G e. T )
16 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
17 14 15 13 16 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( G ` P ) e. A )
18 simp23 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` F ) = ( R ` N ) )
19 simp213 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> X e. T )
20 simp12 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> F e. T )
21 simp13 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> D e. T )
22 simp211 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> N e. T )
23 simp331 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
24 simp333 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` X ) =/= ( R ` D ) )
25 24 necomd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` X ) )
26 23 25 jca 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` X ) ) )
27 simp311 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> F =/= ( _I |` B ) )
28 simp32l 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> X =/= ( _I |` B ) )
29 simp312 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> D =/= ( _I |` B ) )
30 27 28 29 3jca 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) )
31 simp22 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
32 1 2 3 4 5 6 7 8 9 10 11 cdlemkuat 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ X e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` X ) ) /\ ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` X ) ` P ) e. A )
33 14 18 19 20 21 22 26 30 31 32 syl333anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` X ) ` P ) e. A )
34 simp32r 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) =/= ( R ` X ) )
35 34 necomd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` X ) =/= ( R ` G ) )
36 5 6 7 8 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ G e. T ) /\ ( R ` X ) =/= ( R ` G ) ) -> ( R ` ( X o. `' G ) ) e. A )
37 14 19 15 35 36 syl121anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( X o. `' G ) ) e. A )
38 simp332 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
39 38 necomd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` G ) )
40 23 39 jca 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) )
41 simp313 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> G =/= ( _I |` B ) )
42 27 41 29 3jca 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) )
43 1 2 3 4 5 6 7 8 9 10 11 cdlemkuat 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` G ) ` P ) e. A )
44 14 18 15 20 21 22 40 42 31 43 syl333anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) e. A )
45 1 2 3 4 5 6 7 8 9 10 11 cdlemkuv2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
46 14 18 15 20 21 22 40 42 31 45 syl333anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
47 12 hllatd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> K e. Lat )
48 1 5 6 7 8 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
49 14 15 41 48 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) e. A )
50 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( P .\/ ( R ` G ) ) e. B )
51 12 13 49 50 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( R ` G ) ) e. B )
52 simp1 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
53 22 31 18 3jca 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
54 1 2 3 4 5 6 7 8 9 10 cdlemkoatnle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )
55 54 simpld 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) e. A )
56 52 53 27 29 23 55 syl113anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( O ` P ) e. A )
57 5 6 7 8 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ D e. T ) /\ ( R ` G ) =/= ( R ` D ) ) -> ( R ` ( G o. `' D ) ) e. A )
58 14 15 21 38 57 syl121anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( G o. `' D ) ) e. A )
59 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ ( O ` P ) e. A /\ ( R ` ( G o. `' D ) ) e. A ) -> ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B )
60 12 56 58 59 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B )
61 1 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ ( R ` G ) ) e. B /\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( P .\/ ( R ` G ) ) )
62 47 51 60 61 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( P .\/ ( R ` G ) ) )
63 46 62 eqbrtrd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) .<_ ( P .\/ ( R ` G ) ) )
64 2 3 5 6 7 8 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
65 14 15 31 64 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
66 63 65 breqtrd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) .<_ ( P .\/ ( G ` P ) ) )
67 simp2 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
68 simp31 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
69 simp33 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) )
70 eqid 
 |-  ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) ) = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' D ) ) .\/ ( R ` ( X o. `' D ) ) ) )
71 1 2 3 4 5 6 7 8 9 10 11 70 cdlemk11u 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) .<_ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
72 52 67 68 28 69 71 syl113anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) .<_ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
73 2 3 5 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
74 12 13 49 73 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
75 74 65 breqtrd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) .<_ ( P .\/ ( G ` P ) ) )
76 1 2 3 4 5 6 7 8 9 10 11 cdlemkuel 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ X e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` X ) ) /\ ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( U ` X ) e. T )
77 14 18 19 20 21 22 26 30 31 76 syl333anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( U ` X ) e. T )
78 2 5 6 7 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U ` X ) e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( U ` X ) ` P ) e. A /\ -. ( ( U ` X ) ` P ) .<_ W ) )
79 14 77 31 78 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( ( U ` X ) ` P ) e. A /\ -. ( ( U ` X ) ` P ) .<_ W ) )
80 6 7 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> `' G e. T )
81 14 15 80 syl2anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> `' G e. T )
82 6 7 8 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` `' G ) = ( R ` G ) )
83 14 15 82 syl2anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` `' G ) = ( R ` G ) )
84 83 34 eqnetrd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` `' G ) =/= ( R ` X ) )
85 1 6 7 8 trlcone 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' G e. T /\ X e. T ) /\ ( ( R ` `' G ) =/= ( R ` X ) /\ X =/= ( _I |` B ) ) ) -> ( R ` `' G ) =/= ( R ` ( `' G o. X ) ) )
86 14 81 19 84 28 85 syl122anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` `' G ) =/= ( R ` ( `' G o. X ) ) )
87 86 necomd 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( `' G o. X ) ) =/= ( R ` `' G ) )
88 6 7 ltrncom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ `' G e. T /\ X e. T ) -> ( `' G o. X ) = ( X o. `' G ) )
89 14 81 19 88 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( `' G o. X ) = ( X o. `' G ) )
90 89 fveq2d 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( `' G o. X ) ) = ( R ` ( X o. `' G ) ) )
91 87 90 83 3netr3d 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( X o. `' G ) ) =/= ( R ` G ) )
92 6 7 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ `' G e. T ) -> ( X o. `' G ) e. T )
93 14 19 81 92 syl3anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( X o. `' G ) e. T )
94 2 6 7 8 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X o. `' G ) e. T ) -> ( R ` ( X o. `' G ) ) .<_ W )
95 14 93 94 syl2anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( X o. `' G ) ) .<_ W )
96 2 6 7 8 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
97 14 15 96 syl2anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) .<_ W )
98 2 3 5 6 lhp2atnle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( ( U ` X ) ` P ) e. A /\ -. ( ( U ` X ) ` P ) .<_ W ) /\ ( R ` ( X o. `' G ) ) =/= ( R ` G ) ) /\ ( ( R ` ( X o. `' G ) ) e. A /\ ( R ` ( X o. `' G ) ) .<_ W ) /\ ( ( R ` G ) e. A /\ ( R ` G ) .<_ W ) ) -> -. ( R ` G ) .<_ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
99 14 79 91 37 95 49 97 98 syl322anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> -. ( R ` G ) .<_ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
100 nbrne1 
 |-  ( ( ( R ` G ) .<_ ( P .\/ ( G ` P ) ) /\ -. ( R ` G ) .<_ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) -> ( P .\/ ( G ` P ) ) =/= ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
101 75 99 100 syl2anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( G ` P ) ) =/= ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
102 2 3 4 5 2atm 
 |-  ( ( ( K e. HL /\ P e. A /\ ( G ` P ) e. A ) /\ ( ( ( U ` X ) ` P ) e. A /\ ( R ` ( X o. `' G ) ) e. A /\ ( ( U ` G ) ` P ) e. A ) /\ ( ( ( U ` G ) ` P ) .<_ ( P .\/ ( G ` P ) ) /\ ( ( U ` G ) ` P ) .<_ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) /\ ( P .\/ ( G ` P ) ) =/= ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )
103 12 13 17 33 37 44 66 72 101 102 syl333anc 
 |-  ( ( ( ( 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( U ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )