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Metamath Proof Explorer

Theorem cdleme4a

Description: Part of proof of Lemma E in Crawley p. 114 top. G represents f_s(r). Auxiliary lemma derived from cdleme5 . We show f_s(r) <_ p \/ q. (Contributed by NM, 10-Nov-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 ) ) )
Assertion cdleme4a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> G .<_ ( P .\/ Q ) )

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 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> K e. HL )
10 9 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> K e. Lat )
11 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> P e. A )
12 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> Q e. A )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
15 9 11 12 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
16 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> W e. H )
17 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> S e. A )
18 1 2 3 4 5 6 7 13 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
19 9 16 11 12 17 18 syl23anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> F e. ( Base ` K ) )
20 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> R e. A )
21 13 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
22 9 20 17 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
23 13 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
24 16 23 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> W e. ( Base ` K ) )
25 13 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) )
26 10 22 24 25 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) )
27 13 2 latjcl 
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ ( ( R .\/ S ) ./\ W ) e. ( Base ` K ) ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) )
28 10 19 26 27 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) )
29 13 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
30 10 15 28 29 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
31 8 30 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ S e. A ) -> G .<_ ( P .\/ Q ) )