Metamath Proof Explorer

Theorem cdleme35c

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

Ref Expression
Hypotheses cdleme35.l 
|- .<_ = ( le ` K )
cdleme35.j 
|- .\/ = ( join ` K )
cdleme35.m 
|- ./\ = ( meet ` K )
cdleme35.a 
|- A = ( Atoms ` K )
cdleme35.h 
|- H = ( LHyp ` K )
cdleme35.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme35.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdleme35c 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ F ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdleme35.l 
 |-  .<_ = ( le ` K )
2 cdleme35.j 
 |-  .\/ = ( join ` K )
3 cdleme35.m 
 |-  ./\ = ( meet ` K )
4 cdleme35.a 
 |-  A = ( Atoms ` K )
5 cdleme35.h 
 |-  H = ( LHyp ` K )
6 cdleme35.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme35.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 7 oveq2i 
 |-  ( Q .\/ F ) = ( Q .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
9 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. HL )
10 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. A )
11 simp2rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
12 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) )
13 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) )
14 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= Q )
15 1 2 3 4 5 6 cdleme0a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
16 12 13 10 14 15 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> U e. A )
17 eqid 
 |-  ( Base ` K ) = ( Base ` K )
18 17 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R .\/ U ) e. ( Base ` K ) )
19 9 11 16 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .\/ U ) e. ( Base ` K ) )
20 9 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. Lat )
21 17 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
22 10 21 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. ( Base ` K ) )
23 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. A )
24 17 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) )
25 9 23 11 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ R ) e. ( Base ` K ) )
26 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> W e. H )
27 17 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
28 26 27 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> W e. ( Base ` K ) )
29 17 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
30 20 25 28 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
31 17 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
32 20 22 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
33 17 1 2 latlej1 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) ) -> Q .<_ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
34 20 22 30 33 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q .<_ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
35 17 1 2 3 4 atmod1i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( R .\/ U ) e. ( Base ` K ) /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) ) /\ Q .<_ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) -> ( Q .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) = ( ( Q .\/ ( R .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
36 9 10 19 32 34 35 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) = ( ( Q .\/ ( R .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
37 1 2 3 4 5 6 7 cdleme35b 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) )
38 17 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) ) -> ( Q .\/ ( R .\/ U ) ) e. ( Base ` K ) )
39 20 22 19 38 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( R .\/ U ) ) e. ( Base ` K ) )
40 17 1 3 latleeqm2 
 |-  ( ( K e. Lat /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) /\ ( Q .\/ ( R .\/ U ) ) e. ( Base ` K ) ) -> ( ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) <-> ( ( Q .\/ ( R .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
41 20 32 39 40 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) <-> ( ( Q .\/ ( R .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
42 37 41 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .\/ ( R .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
43 36 42 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
44 8 43 eqtrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ F ) = ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )