Metamath Proof Explorer

Theorem cdleme11g

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 14-Jun-2012)

Ref Expression
Hypotheses cdleme11.l 
|- .<_ = ( le ` K )
cdleme11.j 
|- .\/ = ( join ` K )
cdleme11.m 
|- ./\ = ( meet ` K )
cdleme11.a 
|- A = ( Atoms ` K )
cdleme11.h 
|- H = ( LHyp ` K )
cdleme11.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme11.c 
|- C = ( ( P .\/ S ) ./\ W )
cdleme11.d 
|- D = ( ( P .\/ T ) ./\ W )
cdleme11.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
Assertion cdleme11g 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ F ) = ( Q .\/ C ) )

Proof

Step Hyp Ref Expression
1 cdleme11.l 
 |-  .<_ = ( le ` K )
2 cdleme11.j 
 |-  .\/ = ( join ` K )
3 cdleme11.m 
 |-  ./\ = ( meet ` K )
4 cdleme11.a 
 |-  A = ( Atoms ` K )
5 cdleme11.h 
 |-  H = ( LHyp ` K )
6 cdleme11.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme11.c 
 |-  C = ( ( P .\/ S ) ./\ W )
8 cdleme11.d 
 |-  D = ( ( P .\/ T ) ./\ W )
9 cdleme11.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
10 9 oveq2i 
 |-  ( Q .\/ F ) = ( Q .\/ ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
11 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> K e. HL )
12 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> Q e. A )
13 11 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> K e. Lat )
14 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> S e. A )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
17 14 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> S e. ( Base ` K ) )
18 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
19 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> P e. A )
20 1 2 3 4 5 6 15 cdleme0aa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. ( Base ` K ) )
21 18 19 12 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> U e. ( Base ` K ) )
22 15 2 latjcl 
 |-  ( ( K e. Lat /\ S e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( S .\/ U ) e. ( Base ` K ) )
23 13 17 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( S .\/ U ) e. ( Base ` K ) )
24 15 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
25 12 24 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> Q e. ( Base ` K ) )
26 15 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
27 19 26 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> P e. ( Base ` K ) )
28 15 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( P .\/ S ) e. ( Base ` K ) )
29 13 27 17 28 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( P .\/ S ) e. ( Base ` K ) )
30 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> W e. H )
31 15 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
32 30 31 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> W e. ( Base ` K ) )
33 15 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) e. ( Base ` K ) )
34 13 29 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( P .\/ S ) ./\ W ) e. ( Base ` K ) )
35 15 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ W ) e. ( Base ` K ) ) -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) e. ( Base ` K ) )
36 13 25 34 35 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) e. ( Base ` K ) )
37 15 1 2 latlej1 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ W ) e. ( Base ` K ) ) -> Q .<_ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
38 13 25 34 37 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> Q .<_ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
39 15 1 2 3 4 atmod1i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( S .\/ U ) e. ( Base ` K ) /\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) e. ( Base ` K ) ) /\ Q .<_ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) -> ( Q .\/ ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) ) = ( ( Q .\/ ( S .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
40 11 12 23 36 38 39 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) ) = ( ( Q .\/ ( S .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
41 10 40 eqtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ F ) = ( ( Q .\/ ( S .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
42 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q e. A /\ -. Q .<_ W ) )
43 1 2 3 4 5 6 cdleme0cq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( Q .\/ U ) = ( P .\/ Q ) )
44 18 19 42 43 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ U ) = ( P .\/ Q ) )
45 44 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( S .\/ ( Q .\/ U ) ) = ( S .\/ ( P .\/ Q ) ) )
46 15 2 latj12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ S e. ( Base ` K ) /\ U e. ( Base ` K ) ) ) -> ( Q .\/ ( S .\/ U ) ) = ( S .\/ ( Q .\/ U ) ) )
47 13 25 17 21 46 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( S .\/ U ) ) = ( S .\/ ( Q .\/ U ) ) )
48 15 2 latj13 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( Q .\/ ( P .\/ S ) ) = ( S .\/ ( P .\/ Q ) ) )
49 13 25 27 17 48 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( P .\/ S ) ) = ( S .\/ ( P .\/ Q ) ) )
50 45 47 49 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( S .\/ U ) ) = ( Q .\/ ( P .\/ S ) ) )
51 50 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( Q .\/ ( S .\/ U ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
52 15 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
53 13 29 32 52 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
54 15 1 2 latjlej2 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ S ) ./\ W ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) ) -> ( ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) .<_ ( Q .\/ ( P .\/ S ) ) ) )
55 13 34 29 25 54 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) .<_ ( Q .\/ ( P .\/ S ) ) ) )
56 53 55 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) .<_ ( Q .\/ ( P .\/ S ) ) )
57 15 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) )
58 13 25 29 57 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) )
59 15 1 3 latleeqm2 
 |-  ( ( K e. Lat /\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) e. ( Base ` K ) /\ ( Q .\/ ( P .\/ S ) ) e. ( Base ` K ) ) -> ( ( Q .\/ ( ( P .\/ S ) ./\ W ) ) .<_ ( Q .\/ ( P .\/ S ) ) <-> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
60 13 36 58 59 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( Q .\/ ( ( P .\/ S ) ./\ W ) ) .<_ ( Q .\/ ( P .\/ S ) ) <-> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
61 56 60 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
62 7 oveq2i 
 |-  ( Q .\/ C ) = ( Q .\/ ( ( P .\/ S ) ./\ W ) )
63 61 62 eqtr4di 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( ( Q .\/ ( P .\/ S ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( Q .\/ C ) )
64 41 51 63 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ F ) = ( Q .\/ C ) )