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Metamath Proof Explorer

Theorem cdleme0cp

Description: Part of proof of Lemma E in Crawley p. 113. TODO: Reformat as in cdlemg3a - swap consequent equality; make antecedent use df-3an . (Contributed by NM, 13-Jun-2012)

Ref Expression
Hypotheses cdleme0.l 
|- .<_ = ( le ` K )
cdleme0.j 
|- .\/ = ( join ` K )
cdleme0.m 
|- ./\ = ( meet ` K )
cdleme0.a 
|- A = ( Atoms ` K )
cdleme0.h 
|- H = ( LHyp ` K )
cdleme0.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme0cp 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ U ) = ( P .\/ Q ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` K )
2 cdleme0.j 
 |-  .\/ = ( join ` K )
3 cdleme0.m 
 |-  ./\ = ( meet ` K )
4 cdleme0.a 
 |-  A = ( Atoms ` K )
5 cdleme0.h 
 |-  H = ( LHyp ` K )
6 cdleme0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 6 oveq2i 
 |-  ( P .\/ U ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) )
8 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> K e. HL )
9 simprll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> P e. A )
10 hllat 
 |-  ( K e. HL -> K e. Lat )
11 10 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> K e. Lat )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
14 9 13 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> P e. ( Base ` K ) )
15 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> Q e. A )
16 12 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
17 15 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> Q e. ( Base ` K ) )
18 12 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
19 11 14 17 18 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
20 12 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
21 20 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> W e. ( Base ` K ) )
22 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) )
23 8 9 15 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> P .<_ ( P .\/ Q ) )
24 12 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ Q ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( P .\/ W ) ) )
25 8 9 19 21 23 24 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( P .\/ W ) ) )
26 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
27 1 2 26 4 5 lhpjat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = ( 1. ` K ) )
28 27 adantrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ W ) = ( 1. ` K ) )
29 28 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( ( P .\/ Q ) ./\ ( P .\/ W ) ) = ( ( P .\/ Q ) ./\ ( 1. ` K ) ) )
30 hlol 
 |-  ( K e. HL -> K e. OL )
31 30 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> K e. OL )
32 12 3 26 olm11 
 |-  ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
33 31 19 32 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
34 25 29 33 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) )
35 7 34 eqtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ U ) = ( P .\/ Q ) )