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Metamath Proof Explorer

Theorem cdlemd6

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 31-May-2012)

Ref Expression
Hypotheses cdlemd4.l 
|- .<_ = ( le ` K )
cdlemd4.j 
|- .\/ = ( join ` K )
cdlemd4.a 
|- A = ( Atoms ` K )
cdlemd4.h 
|- H = ( LHyp ` K )
cdlemd4.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemd6 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( F ` Q ) = ( G ` Q ) )

Proof

Step Hyp Ref Expression
1 cdlemd4.l 
 |-  .<_ = ( le ` K )
2 cdlemd4.j 
 |-  .\/ = ( join ` K )
3 cdlemd4.a 
 |-  A = ( Atoms ` K )
4 cdlemd4.h 
 |-  H = ( LHyp ` K )
5 cdlemd4.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( F ` P ) = ( G ` P ) )
7 6 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( P .\/ ( F ` P ) ) = ( P .\/ ( G ` P ) ) )
8 7 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) )
9 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( K e. HL /\ W e. H ) )
10 simp1rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> F e. T )
11 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( P e. A /\ -. P .<_ W ) )
12 eqid 
 |-  ( meet ` K ) = ( meet ` K )
13 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
14 1 2 12 3 4 5 13 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` F ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) )
15 9 10 11 14 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( ( ( trL ` K ) ` W ) ` F ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) )
16 simp1rr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> G e. T )
17 1 2 12 3 4 5 13 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` G ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) )
18 9 16 11 17 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( ( ( trL ` K ) ` W ) ` G ) = ( ( P .\/ ( G ` P ) ) ( meet ` K ) W ) )
19 8 15 18 3eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( ( ( trL ` K ) ` W ) ` F ) = ( ( ( trL ` K ) ` W ) ` G ) )
20 19 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( Q .\/ ( ( ( trL ` K ) ` W ) ` F ) ) = ( Q .\/ ( ( ( trL ` K ) ` W ) ` G ) ) )
21 6 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( ( F ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) = ( ( G ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) )
22 20 21 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( ( Q .\/ ( ( ( trL ` K ) ` W ) ` F ) ) ( meet ` K ) ( ( F ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) ) = ( ( Q .\/ ( ( ( trL ` K ) ` W ) ` G ) ) ( meet ` K ) ( ( G ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) ) )
23 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( Q e. A /\ -. Q .<_ W ) )
24 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> -. Q .<_ ( P .\/ ( F ` P ) ) )
25 1 2 12 3 4 5 13 cdlemc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) -> ( F ` Q ) = ( ( Q .\/ ( ( ( trL ` K ) ` W ) ` F ) ) ( meet ` K ) ( ( F ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) ) )
26 9 10 11 23 24 25 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( F ` Q ) = ( ( Q .\/ ( ( ( trL ` K ) ` W ) ` F ) ) ( meet ` K ) ( ( F ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) ) )
27 oveq2 
 |-  ( ( F ` P ) = ( G ` P ) -> ( P .\/ ( F ` P ) ) = ( P .\/ ( G ` P ) ) )
28 27 breq2d 
 |-  ( ( F ` P ) = ( G ` P ) -> ( Q .<_ ( P .\/ ( F ` P ) ) <-> Q .<_ ( P .\/ ( G ` P ) ) ) )
29 28 notbid 
 |-  ( ( F ` P ) = ( G ` P ) -> ( -. Q .<_ ( P .\/ ( F ` P ) ) <-> -. Q .<_ ( P .\/ ( G ` P ) ) ) )
30 29 biimpd 
 |-  ( ( F ` P ) = ( G ` P ) -> ( -. Q .<_ ( P .\/ ( F ` P ) ) -> -. Q .<_ ( P .\/ ( G ` P ) ) ) )
31 6 24 30 sylc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> -. Q .<_ ( P .\/ ( G ` P ) ) )
32 1 2 12 3 4 5 13 cdlemc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ -. Q .<_ ( P .\/ ( G ` P ) ) ) -> ( G ` Q ) = ( ( Q .\/ ( ( ( trL ` K ) ` W ) ` G ) ) ( meet ` K ) ( ( G ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) ) )
33 9 16 11 23 31 32 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( G ` Q ) = ( ( Q .\/ ( ( ( trL ` K ) ` W ) ` G ) ) ( meet ` K ) ( ( G ` P ) .\/ ( ( P .\/ Q ) ( meet ` K ) W ) ) ) )
34 22 26 33 3eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ -. Q .<_ ( P .\/ ( F ` P ) ) ) /\ ( F ` P ) = ( G ` P ) ) -> ( F ` Q ) = ( G ` Q ) )