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Metamath Proof Explorer

Theorem cdleme35f

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 cdleme35f 
|- ( ( ( ( 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 ) ./\ ( P .\/ R ) ) = R )

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 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 )
9 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 )
10 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 )
11 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) = ( R .\/ P ) )
12 8 9 10 11 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 ) = ( R .\/ P ) )
13 12 oveq2d 
 |-  ( ( ( ( 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 ) ./\ ( P .\/ R ) ) = ( ( R .\/ U ) ./\ ( R .\/ P ) ) )
14 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 ) )
15 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 ) )
16 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 )
17 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 )
18 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 )
19 14 15 16 17 18 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 )
20 simp12r 
 |-  ( ( ( ( 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 .<_ W )
21 8 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 )
22 eqid 
 |-  ( Base ` K ) = ( Base ` K )
23 22 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
24 8 9 16 23 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 .\/ Q ) e. ( Base ` K ) )
25 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 )
26 22 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
27 25 26 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 ) )
28 22 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
29 21 24 27 28 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 .\/ Q ) ./\ W ) .<_ W )
30 6 29 eqbrtrid 
 |-  ( ( ( ( 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 .<_ W )
31 breq1 
 |-  ( U = P -> ( U .<_ W <-> P .<_ W ) )
32 30 31 syl5ibcom 
 |-  ( ( ( ( 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 = P -> P .<_ W ) )
33 32 necon3bd 
 |-  ( ( ( ( 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 .<_ W -> U =/= P ) )
34 20 33 mpd 
 |-  ( ( ( ( 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 =/= P )
35 simp3 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) )
36 22 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
37 21 24 27 36 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 .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
38 6 37 eqbrtrid 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) )
39 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) )
40 8 9 16 39 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 .<_ ( P .\/ Q ) )
41 22 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
42 19 41 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 ) ) -> U e. ( Base ` K ) )
43 22 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
44 9 43 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 ) ) -> P e. ( Base ` K ) )
45 22 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( U .<_ ( P .\/ Q ) /\ P .<_ ( P .\/ Q ) ) <-> ( U .\/ P ) .<_ ( P .\/ Q ) ) )
46 21 42 44 24 45 syl13anc 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) /\ P .<_ ( P .\/ Q ) ) <-> ( U .\/ P ) .<_ ( P .\/ Q ) ) )
47 38 40 46 mpbi2and 
 |-  ( ( ( ( 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 .\/ P ) .<_ ( P .\/ Q ) )
48 22 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
49 10 48 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 ) ) -> R e. ( Base ` K ) )
50 22 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ P e. A ) -> ( U .\/ P ) e. ( Base ` K ) )
51 8 19 9 50 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 ) ) -> ( U .\/ P ) e. ( Base ` K ) )
52 22 1 lattr 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ ( U .\/ P ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( R .<_ ( U .\/ P ) /\ ( U .\/ P ) .<_ ( P .\/ Q ) ) -> R .<_ ( P .\/ Q ) ) )
53 21 49 51 24 52 syl13anc 
 |-  ( ( ( ( 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 .\/ P ) /\ ( U .\/ P ) .<_ ( P .\/ Q ) ) -> R .<_ ( P .\/ Q ) ) )
54 47 53 mpan2d 
 |-  ( ( ( ( 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 .\/ P ) -> R .<_ ( P .\/ Q ) ) )
55 35 54 mtod 
 |-  ( ( ( ( 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 .\/ P ) )
56 1 2 3 4 2llnma2 
 |-  ( ( K e. HL /\ ( U e. A /\ P e. A /\ R e. A ) /\ ( U =/= P /\ -. R .<_ ( U .\/ P ) ) ) -> ( ( R .\/ U ) ./\ ( R .\/ P ) ) = R )
57 8 19 9 10 34 55 56 syl132anc 
 |-  ( ( ( ( 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 ) ./\ ( R .\/ P ) ) = R )
58 13 57 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 ) ) -> ( ( R .\/ U ) ./\ ( P .\/ R ) ) = R )