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Metamath Proof Explorer

Theorem cdlemk6

Description: Part of proof of Lemma K of Crawley p. 118. Apply dalaw . (Contributed by NM, 25-Jun-2013)

Ref Expression
Hypotheses cdlemk.b 
|- B = ( Base ` K )
cdlemk.l 
|- .<_ = ( le ` K )
cdlemk.j 
|- .\/ = ( join ` K )
cdlemk.a 
|- A = ( Atoms ` K )
cdlemk.h 
|- H = ( LHyp ` K )
cdlemk.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk.r 
|- R = ( ( trL ` K ) ` W )
cdlemk.m 
|- ./\ = ( meet ` K )
Assertion cdlemk6 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk.b 
 |-  B = ( Base ` K )
2 cdlemk.l 
 |-  .<_ = ( le ` K )
3 cdlemk.j 
 |-  .\/ = ( join ` K )
4 cdlemk.a 
 |-  A = ( Atoms ` K )
5 cdlemk.h 
 |-  H = ( LHyp ` K )
6 cdlemk.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemk.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemk.m 
 |-  ./\ = ( meet ` K )
9 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> F =/= ( _I |` B ) )
10 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> G =/= ( _I |` B ) )
11 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` F ) )
12 9 10 11 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) )
13 1 2 3 4 5 6 7 8 cdlemk5 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
14 12 13 syld3an3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) )
15 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> K e. HL )
16 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> P e. A )
17 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( K e. HL /\ W e. H ) )
18 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> G e. T )
19 2 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
20 17 18 16 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( G ` P ) e. A )
21 simp21r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> X e. T )
22 2 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A )
23 17 21 16 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( X ` P ) e. A )
24 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> N e. T )
25 2 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
26 17 24 16 25 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( N ` P ) e. A )
27 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> F e. T )
28 4 5 6 7 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ F e. T ) /\ ( R ` G ) =/= ( R ` F ) ) -> ( R ` ( G o. `' F ) ) e. A )
29 17 18 27 11 28 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( R ` ( G o. `' F ) ) e. A )
30 simp33r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( R ` X ) =/= ( R ` F ) )
31 4 5 6 7 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ F e. T ) /\ ( R ` X ) =/= ( R ` F ) ) -> ( R ` ( X o. `' F ) ) e. A )
32 17 21 27 30 31 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( R ` ( X o. `' F ) ) e. A )
33 2 3 8 4 dalaw 
 |-  ( ( K e. HL /\ ( P e. A /\ ( G ` P ) e. A /\ ( X ` P ) e. A ) /\ ( ( N ` P ) e. A /\ ( R ` ( G o. `' F ) ) e. A /\ ( R ` ( X o. `' F ) ) e. A ) ) -> ( ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) ) )
34 15 16 20 23 26 29 32 33 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( ( ( P .\/ ( N ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' F ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) ) )
35 14 34 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) )