Metamath Proof Explorer

Theorem cdlemk1

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 22-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 )
Assertion cdlemk1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P .\/ ( N ` P ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )

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 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) )
9 8 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( R ` N ) ) )
10 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
12 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
13 2 3 4 5 6 7 trljat3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` F ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )
14 10 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P .\/ ( R ` F ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )
15 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T )
16 2 3 4 5 6 7 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` N ) ) = ( P .\/ ( N ` P ) ) )
17 10 15 12 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P .\/ ( R ` N ) ) = ( P .\/ ( N ` P ) ) )
18 9 14 17 3eqtr3rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( ( R ` F ) = ( R ` N ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P .\/ ( N ` P ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )