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Metamath Proof Explorer

Theorem cdleme15

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, showing, in their notation, (s \/ t) /\ (f(s) \/ f(t)) <_ w. We use F , G for f(s), f(t) respectively. (Contributed by NM, 10-Oct-2012)

Ref Expression
Hypotheses cdleme12.l 
|- .<_ = ( le ` K )
cdleme12.j 
|- .\/ = ( join ` K )
cdleme12.m 
|- ./\ = ( meet ` K )
cdleme12.a 
|- A = ( Atoms ` K )
cdleme12.h 
|- H = ( LHyp ` K )
cdleme12.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme12.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme12.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
Assertion cdleme15 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ W )

Proof

Step Hyp Ref Expression
1 cdleme12.l 
 |-  .<_ = ( le ` K )
2 cdleme12.j 
 |-  .\/ = ( join ` K )
3 cdleme12.m 
 |-  ./\ = ( meet ` K )
4 cdleme12.a 
 |-  A = ( Atoms ` K )
5 cdleme12.h 
 |-  H = ( LHyp ` K )
6 cdleme12.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme12.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme12.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> K e. Lat )
12 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> S e. A )
13 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> T e. A )
14 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
15 10 12 13 14 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
16 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> W e. H )
17 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> P e. A )
18 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> Q e. A )
19 1 2 3 4 5 6 7 9 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
20 10 16 17 18 12 19 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> F e. ( Base ` K ) )
21 1 2 3 4 5 6 8 9 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ T e. A ) ) -> G e. ( Base ` K ) )
22 10 16 17 18 13 21 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> G e. ( Base ` K ) )
23 9 2 latjcl 
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ G e. ( Base ` K ) ) -> ( F .\/ G ) e. ( Base ` K ) )
24 11 20 22 23 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( F .\/ G ) e. ( Base ` K ) )
25 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ ( F .\/ G ) e. ( Base ` K ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) e. ( Base ` K ) )
26 11 15 24 25 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) e. ( Base ` K ) )
27 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ P e. A ) -> ( T .\/ P ) e. ( Base ` K ) )
28 10 13 17 27 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( T .\/ P ) e. ( Base ` K ) )
29 9 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
30 18 29 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> Q e. ( Base ` K ) )
31 9 2 latjcl 
 |-  ( ( K e. Lat /\ G e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( G .\/ Q ) e. ( Base ` K ) )
32 11 22 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( G .\/ Q ) e. ( Base ` K ) )
33 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( T .\/ P ) e. ( Base ` K ) /\ ( G .\/ Q ) e. ( Base ` K ) ) -> ( ( T .\/ P ) ./\ ( G .\/ Q ) ) e. ( Base ` K ) )
34 11 28 32 33 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( T .\/ P ) ./\ ( G .\/ Q ) ) e. ( Base ` K ) )
35 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) )
36 10 17 12 35 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( P .\/ S ) e. ( Base ` K ) )
37 9 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ F e. ( Base ` K ) ) -> ( Q .\/ F ) e. ( Base ` K ) )
38 11 30 20 37 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( Q .\/ F ) e. ( Base ` K ) )
39 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ ( Q .\/ F ) e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ F ) ) e. ( Base ` K ) )
40 11 36 38 39 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ F ) ) e. ( Base ` K ) )
41 9 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( T .\/ P ) ./\ ( G .\/ Q ) ) e. ( Base ` K ) /\ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) e. ( Base ` K ) ) -> ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) e. ( Base ` K ) )
42 11 34 40 41 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) e. ( Base ` K ) )
43 9 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
44 16 43 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> W e. ( Base ` K ) )
45 1 2 3 4 5 6 7 8 cdleme14 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) )
46 eqid 
 |-  ( ( P .\/ S ) ./\ W ) = ( ( P .\/ S ) ./\ W )
47 eqid 
 |-  ( ( P .\/ T ) ./\ W ) = ( ( P .\/ T ) ./\ W )
48 1 2 3 4 5 6 7 8 46 47 cdleme15a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) = ( ( ( P .\/ ( ( P .\/ T ) ./\ W ) ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) .\/ ( ( P .\/ ( ( P .\/ S ) ./\ W ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) ) )
49 1 2 3 4 5 6 7 8 46 47 cdleme15c 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( P .\/ ( ( P .\/ T ) ./\ W ) ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) .\/ ( ( P .\/ ( ( P .\/ S ) ./\ W ) ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) ) = ( ( ( P .\/ T ) ./\ W ) .\/ ( ( P .\/ S ) ./\ W ) ) )
50 48 49 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) = ( ( ( P .\/ T ) ./\ W ) .\/ ( ( P .\/ S ) ./\ W ) ) )
51 1 2 3 4 5 6 7 8 46 47 cdleme15d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( P .\/ T ) ./\ W ) .\/ ( ( P .\/ S ) ./\ W ) ) .<_ W )
52 50 51 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( ( T .\/ P ) ./\ ( G .\/ Q ) ) .\/ ( ( P .\/ S ) ./\ ( Q .\/ F ) ) ) .<_ W )
53 9 1 11 26 42 44 45 52 lattrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) .<_ W )