Metamath Proof Explorer

Theorem cdleme1

Description: Part of proof of Lemma E in Crawley p. 113. F represents their f(r). Here we show r \/ f(r) = r \/ u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-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 ) ) )
Assertion cdleme1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )

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 7 oveq2i 
 |-  ( R .\/ F ) = ( R .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
9 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
10 simpr3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
11 hllat 
 |-  ( K e. HL -> K e. Lat )
12 11 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. Lat )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
15 10 14 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. ( Base ` K ) )
16 simpr1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
17 13 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
18 16 17 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. ( Base ` K ) )
19 simpr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. A )
20 13 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
21 19 20 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. ( Base ` K ) )
22 13 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
23 12 18 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
24 13 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
25 24 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> W e. ( Base ` K ) )
26 13 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
27 12 23 25 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
28 6 27 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. ( Base ` K ) )
29 13 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( R .\/ U ) e. ( Base ` K ) )
30 12 15 28 29 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ U ) e. ( Base ` K ) )
31 13 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) e. ( Base ` K ) )
32 12 18 15 31 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ R ) e. ( Base ` K ) )
33 13 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
34 12 32 25 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
35 13 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
36 12 21 34 35 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
37 13 1 2 latlej1 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> R .<_ ( R .\/ U ) )
38 12 15 28 37 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> R .<_ ( R .\/ U ) )
39 13 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( R e. A /\ ( R .\/ U ) e. ( Base ` K ) /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) ) /\ R .<_ ( R .\/ U ) ) -> ( R .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) = ( ( R .\/ U ) ./\ ( R .\/ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) )
40 9 10 30 36 38 39 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) = ( ( R .\/ U ) ./\ ( R .\/ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) )
41 13 1 2 latlej2 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> R .<_ ( P .\/ R ) )
42 12 18 15 41 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> R .<_ ( P .\/ R ) )
43 13 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( R e. A /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ R .<_ ( P .\/ R ) ) -> ( R .\/ ( ( P .\/ R ) ./\ W ) ) = ( ( P .\/ R ) ./\ ( R .\/ W ) ) )
44 9 10 32 25 42 43 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( ( P .\/ R ) ./\ W ) ) = ( ( P .\/ R ) ./\ ( R .\/ W ) ) )
45 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
46 1 2 45 4 5 lhpjat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R .\/ W ) = ( 1. ` K ) )
47 46 3ad2antr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ W ) = ( 1. ` K ) )
48 47 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ R ) ./\ ( R .\/ W ) ) = ( ( P .\/ R ) ./\ ( 1. ` K ) ) )
49 hlol 
 |-  ( K e. HL -> K e. OL )
50 49 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. OL )
51 13 3 45 olm11 
 |-  ( ( K e. OL /\ ( P .\/ R ) e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ ( 1. ` K ) ) = ( P .\/ R ) )
52 50 32 51 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ R ) ./\ ( 1. ` K ) ) = ( P .\/ R ) )
53 44 48 52 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( ( P .\/ R ) ./\ W ) ) = ( P .\/ R ) )
54 53 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( Q .\/ ( R .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( Q .\/ ( P .\/ R ) ) )
55 13 2 latj12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ R e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) ) ) -> ( Q .\/ ( R .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( R .\/ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
56 12 21 15 34 55 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( Q .\/ ( R .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( R .\/ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
57 13 2 latj13 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) ) -> ( Q .\/ ( P .\/ R ) ) = ( R .\/ ( P .\/ Q ) ) )
58 12 21 18 15 57 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( Q .\/ ( P .\/ R ) ) = ( R .\/ ( P .\/ Q ) ) )
59 54 56 58 3eqtr3rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( P .\/ Q ) ) = ( R .\/ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
60 59 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ U ) ./\ ( R .\/ ( P .\/ Q ) ) ) = ( ( R .\/ U ) ./\ ( R .\/ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) )
61 1 2 3 4 5 6 cdlemeulpq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )
62 61 3adantr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> U .<_ ( P .\/ Q ) )
63 13 1 2 latjlej2 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) ) -> ( U .<_ ( P .\/ Q ) -> ( R .\/ U ) .<_ ( R .\/ ( P .\/ Q ) ) ) )
64 12 28 23 15 63 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( U .<_ ( P .\/ Q ) -> ( R .\/ U ) .<_ ( R .\/ ( P .\/ Q ) ) ) )
65 62 64 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ U ) .<_ ( R .\/ ( P .\/ Q ) ) )
66 13 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( R .\/ ( P .\/ Q ) ) e. ( Base ` K ) )
67 12 15 23 66 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( P .\/ Q ) ) e. ( Base ` K ) )
68 13 1 3 latleeqm1 
 |-  ( ( K e. Lat /\ ( R .\/ U ) e. ( Base ` K ) /\ ( R .\/ ( P .\/ Q ) ) e. ( Base ` K ) ) -> ( ( R .\/ U ) .<_ ( R .\/ ( P .\/ Q ) ) <-> ( ( R .\/ U ) ./\ ( R .\/ ( P .\/ Q ) ) ) = ( R .\/ U ) ) )
69 12 30 67 68 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ U ) .<_ ( R .\/ ( P .\/ Q ) ) <-> ( ( R .\/ U ) ./\ ( R .\/ ( P .\/ Q ) ) ) = ( R .\/ U ) ) )
70 65 69 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ U ) ./\ ( R .\/ ( P .\/ Q ) ) ) = ( R .\/ U ) )
71 40 60 70 3eqtr2rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ U ) = ( R .\/ ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) ) )
72 8 71 eqtr4id 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )