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Metamath Proof Explorer

Theorem cdlemeda

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. D represents s_2. (Contributed by NM, 13-Nov-2012)

Ref Expression
Hypotheses cdlemeda.l 
|- .<_ = ( le ` K )
cdlemeda.j 
|- .\/ = ( join ` K )
cdlemeda.m 
|- ./\ = ( meet ` K )
cdlemeda.a 
|- A = ( Atoms ` K )
cdlemeda.h 
|- H = ( LHyp ` K )
cdlemeda.d 
|- D = ( ( R .\/ S ) ./\ W )
Assertion cdlemeda 
|- ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D e. A )

Proof

Step Hyp Ref Expression
1 cdlemeda.l 
 |-  .<_ = ( le ` K )
2 cdlemeda.j 
 |-  .\/ = ( join ` K )
3 cdlemeda.m 
 |-  ./\ = ( meet ` K )
4 cdlemeda.a 
 |-  A = ( Atoms ` K )
5 cdlemeda.h 
 |-  H = ( LHyp ` K )
6 cdlemeda.d 
 |-  D = ( ( R .\/ S ) ./\ W )
7 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
8 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
9 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
10 2 4 hlatjcom 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) = ( S .\/ R ) )
11 7 8 9 10 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) = ( S .\/ R ) )
12 11 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) = ( ( S .\/ R ) ./\ W ) )
13 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
14 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ W )
15 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
16 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
17 1 2 4 5 cdlemesner 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S =/= R )
18 7 8 9 15 16 17 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S =/= R )
19 1 2 3 4 5 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ S =/= R ) ) -> ( ( S .\/ R ) ./\ W ) e. A )
20 7 13 9 14 8 18 19 syl222anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ R ) ./\ W ) e. A )
21 12 20 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. A )
22 6 21 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D e. A )