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Metamath Proof Explorer

Theorem cdleme3b

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . (Contributed by NM, 6-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 cdleme3b 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= R )

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 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
9 simpr3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. ( Base ` K ) )
13 hllat 
 |-  ( K e. HL -> K e. Lat )
14 13 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. Lat )
15 1 2 3 4 5 6 lhpat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
16 15 3adant3r3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. A )
17 10 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
18 16 17 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. ( Base ` K ) )
19 10 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( R .\/ U ) e. ( Base ` K ) )
20 14 12 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ U ) e. ( Base ` K ) )
21 simpr2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. A )
22 10 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. ( Base ` K ) )
24 simpr1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
25 10 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
26 24 25 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. ( Base ` K ) )
27 10 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) e. ( Base ` K ) )
28 14 26 12 27 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ R ) e. ( Base ` K ) )
29 10 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
30 29 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> W e. ( Base ` K ) )
31 10 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
32 14 28 30 31 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
33 10 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
34 14 23 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
35 10 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ U ) e. ( Base ` K ) /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. ( Base ` K ) )
36 14 20 34 35 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. ( Base ` K ) )
37 7 36 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F e. ( Base ` K ) )
38 10 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ F e. ( Base ` K ) ) -> ( R .\/ F ) e. ( Base ` K ) )
39 14 12 37 38 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) e. ( Base ` K ) )
40 10 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
41 14 26 23 40 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
42 10 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
43 14 41 30 42 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
44 6 43 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U .<_ W )
45 simpr3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> -. R .<_ W )
46 nbrne2 
 |-  ( ( U .<_ W /\ -. R .<_ W ) -> U =/= R )
47 44 45 46 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U =/= R )
48 47 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R =/= U )
49 eqid 
 |-  (
50 2 49 4 atcvr1 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R =/= U <-> R (
51 8 9 16 50 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R =/= U <-> R (
52 48 51 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R (
53 simpr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
54 24 21 53 3jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) )
55 1 2 3 4 5 6 7 cdleme1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
56 54 55 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
57 52 56 breqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R (
58 10 49 cvrne 
 |-  ( ( ( K e. HL /\ R e. ( Base ` K ) /\ ( R .\/ F ) e. ( Base ` K ) ) /\ R (  R =/= ( R .\/ F ) )
59 8 12 39 57 58 syl31anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R =/= ( R .\/ F ) )
60 oveq2 
 |-  ( F = R -> ( R .\/ F ) = ( R .\/ R ) )
61 60 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) /\ F = R ) -> ( R .\/ F ) = ( R .\/ R ) )
62 2 4 hlatjidm 
 |-  ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
63 8 9 62 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ R ) = R )
64 63 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) /\ F = R ) -> ( R .\/ R ) = R )
65 61 64 eqtr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) /\ F = R ) -> R = ( R .\/ F ) )
66 65 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( F = R -> R = ( R .\/ F ) ) )
67 66 necon3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R =/= ( R .\/ F ) -> F =/= R ) )
68 59 67 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= R )