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Metamath Proof Explorer

Theorem cdleme1b

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma showing F is a lattice element. F represents their f(r). (Contributed by NM, 6-Jun-2012)

Ref Expression
Hypotheses cdleme1.l 
|- .<_ = ( le ` K )
cdleme1.j 
|- .\/ = ( join ` K )
cdleme1.m 
|- ./\ = ( meet ` K )
cdleme1.a 
|- A = ( Atoms ` K )
cdleme1.h 
|- H = ( LHyp ` K )
cdleme1.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
cdleme1.b 
|- B = ( Base ` K )
Assertion cdleme1b 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> F e. B )

Proof

Step Hyp Ref Expression
1 cdleme1.l 
 |-  .<_ = ( le ` K )
2 cdleme1.j 
 |-  .\/ = ( join ` K )
3 cdleme1.m 
 |-  ./\ = ( meet ` K )
4 cdleme1.a 
 |-  A = ( Atoms ` K )
5 cdleme1.h 
 |-  H = ( LHyp ` K )
6 cdleme1.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme1.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 cdleme1.b 
 |-  B = ( Base ` K )
9 hllat 
 |-  ( K e. HL -> K e. Lat )
10 9 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. Lat )
11 simpr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
12 8 4 atbase 
 |-  ( R e. A -> R e. B )
13 11 12 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. B )
14 1 2 3 4 5 6 8 cdleme0aa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. B )
15 14 3adant3r3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> U e. B )
16 8 2 latjcl 
 |-  ( ( K e. Lat /\ R e. B /\ U e. B ) -> ( R .\/ U ) e. B )
17 10 13 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( R .\/ U ) e. B )
18 simpr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A )
19 8 4 atbase 
 |-  ( Q e. A -> Q e. B )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. B )
21 simpr1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
22 8 4 atbase 
 |-  ( P e. A -> P e. B )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. B )
24 8 2 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ R e. B ) -> ( P .\/ R ) e. B )
25 10 23 13 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ R ) e. B )
26 8 5 lhpbase 
 |-  ( W e. H -> W e. B )
27 26 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> W e. B )
28 8 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. B /\ W e. B ) -> ( ( P .\/ R ) ./\ W ) e. B )
29 10 25 27 28 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ R ) ./\ W ) e. B )
30 8 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. B /\ ( ( P .\/ R ) ./\ W ) e. B ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. B )
31 10 20 29 30 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. B )
32 8 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ U ) e. B /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. B ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. B )
33 10 17 31 32 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. B )
34 7 33 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> F e. B )