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Metamath Proof Explorer

Theorem cdleme42a

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 3-Mar-2013)

Ref Expression
Hypotheses cdleme42.b 
|- B = ( Base ` K )
cdleme42.l 
|- .<_ = ( le ` K )
cdleme42.j 
|- .\/ = ( join ` K )
cdleme42.m 
|- ./\ = ( meet ` K )
cdleme42.a 
|- A = ( Atoms ` K )
cdleme42.h 
|- H = ( LHyp ` K )
cdleme42.v 
|- V = ( ( R .\/ S ) ./\ W )
Assertion cdleme42a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) = ( R .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdleme42.b 
 |-  B = ( Base ` K )
2 cdleme42.l 
 |-  .<_ = ( le ` K )
3 cdleme42.j 
 |-  .\/ = ( join ` K )
4 cdleme42.m 
 |-  ./\ = ( meet ` K )
5 cdleme42.a 
 |-  A = ( Atoms ` K )
6 cdleme42.h 
 |-  H = ( LHyp ` K )
7 cdleme42.v 
 |-  V = ( ( R .\/ S ) ./\ W )
8 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
9 2 3 8 5 6 lhpjat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
10 9 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
11 10 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ ( R .\/ W ) ) = ( ( R .\/ S ) ./\ ( 1. ` K ) ) )
12 7 oveq2i 
 |-  ( R .\/ V ) = ( R .\/ ( ( R .\/ S ) ./\ W ) )
13 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> K e. HL )
14 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> R e. A )
15 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> S e. A )
16 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. B )
17 13 14 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) e. B )
18 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> W e. H )
19 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> W e. B )
21 2 3 5 hlatlej1 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> R .<_ ( R .\/ S ) )
22 13 14 15 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> R .<_ ( R .\/ S ) )
23 1 2 3 4 5 atmod3i1 
 |-  ( ( K e. HL /\ ( R e. A /\ ( R .\/ S ) e. B /\ W e. B ) /\ R .<_ ( R .\/ S ) ) -> ( R .\/ ( ( R .\/ S ) ./\ W ) ) = ( ( R .\/ S ) ./\ ( R .\/ W ) ) )
24 13 14 17 20 22 23 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ ( ( R .\/ S ) ./\ W ) ) = ( ( R .\/ S ) ./\ ( R .\/ W ) ) )
25 12 24 eqtr2id 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ ( R .\/ W ) ) = ( R .\/ V ) )
26 hlol 
 |-  ( K e. HL -> K e. OL )
27 13 26 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> K e. OL )
28 1 4 8 olm11 
 |-  ( ( K e. OL /\ ( R .\/ S ) e. B ) -> ( ( R .\/ S ) ./\ ( 1. ` K ) ) = ( R .\/ S ) )
29 27 17 28 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ ( 1. ` K ) ) = ( R .\/ S ) )
30 11 25 29 3eqtr3rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) = ( R .\/ V ) )