Metamath Proof Explorer

Theorem cdleme7b

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme7ga and cdleme7 . (Contributed by NM, 7-Jun-2012)

Ref Expression
Hypotheses cdleme4.l 
|- .<_ = ( le ` K )
cdleme4.j 
|- .\/ = ( join ` K )
cdleme4.m 
|- ./\ = ( meet ` K )
cdleme4.a 
|- A = ( Atoms ` K )
cdleme4.h 
|- H = ( LHyp ` K )
cdleme4.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme4.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme4.g 
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
cdleme7.v 
|- V = ( ( R .\/ S ) ./\ W )
Assertion cdleme7b 
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> V e. A )

Proof

Step Hyp Ref Expression
1 cdleme4.l 
 |-  .<_ = ( le ` K )
2 cdleme4.j 
 |-  .\/ = ( join ` K )
3 cdleme4.m 
 |-  ./\ = ( meet ` K )
4 cdleme4.a 
 |-  A = ( Atoms ` K )
5 cdleme4.h 
 |-  H = ( LHyp ` K )
6 cdleme4.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme4.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme4.g 
 |-  G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
9 cdleme7.v 
 |-  V = ( ( R .\/ S ) ./\ W )
10 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
12 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
13 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
14 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
15 nbrne2 
 |-  ( ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) -> R =/= S )
16 13 14 15 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R =/= S )
17 1 2 3 4 5 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ R =/= S ) ) -> ( ( R .\/ S ) ./\ W ) e. A )
18 10 11 12 16 17 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. A )
19 9 18 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> V e. A )