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Metamath Proof Explorer

Theorem cdleme23b

Description: Part of proof of Lemma E in Crawley p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012)

Ref Expression
Hypotheses cdleme23.b 
|- B = ( Base ` K )
cdleme23.l 
|- .<_ = ( le ` K )
cdleme23.j 
|- .\/ = ( join ` K )
cdleme23.m 
|- ./\ = ( meet ` K )
cdleme23.a 
|- A = ( Atoms ` K )
cdleme23.h 
|- H = ( LHyp ` K )
cdleme23.v 
|- V = ( ( S .\/ T ) ./\ ( X ./\ W ) )
Assertion cdleme23b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> V e. A )

Proof

Step Hyp Ref Expression
1 cdleme23.b 
 |-  B = ( Base ` K )
2 cdleme23.l 
 |-  .<_ = ( le ` K )
3 cdleme23.j 
 |-  .\/ = ( join ` K )
4 cdleme23.m 
 |-  ./\ = ( meet ` K )
5 cdleme23.a 
 |-  A = ( Atoms ` K )
6 cdleme23.h 
 |-  H = ( LHyp ` K )
7 cdleme23.v 
 |-  V = ( ( S .\/ T ) ./\ ( X ./\ W ) )
8 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> K e. HL )
9 hlol 
 |-  ( K e. HL -> K e. OL )
10 8 9 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> K e. OL )
11 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> S e. A )
12 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> T e. A )
13 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. B )
14 8 11 12 13 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( S .\/ T ) e. B )
15 8 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> K e. Lat )
16 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> X e. B )
17 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> W e. H )
18 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
19 17 18 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> W e. B )
20 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
21 15 16 19 20 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( X ./\ W ) e. B )
22 1 3 latjcl 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B )
23 15 14 21 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B )
24 1 4 latmassOLD 
 |-  ( ( K e. OL /\ ( ( S .\/ T ) e. B /\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B /\ W e. B ) ) -> ( ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) ./\ W ) = ( ( S .\/ T ) ./\ ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) ) )
25 10 14 23 19 24 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) ./\ W ) = ( ( S .\/ T ) ./\ ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) ) )
26 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) )
27 15 14 21 26 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) )
28 1 2 4 latleeqm1 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B ) -> ( ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) <-> ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) = ( S .\/ T ) ) )
29 15 14 23 28 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) <-> ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) = ( S .\/ T ) ) )
30 27 29 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) = ( S .\/ T ) )
31 30 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) ./\ W ) = ( ( S .\/ T ) ./\ W ) )
32 1 5 atbase 
 |-  ( S e. A -> S e. B )
33 11 32 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> S e. B )
34 1 5 atbase 
 |-  ( T e. A -> T e. B )
35 12 34 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> T e. B )
36 1 3 latjjdir 
 |-  ( ( K e. Lat /\ ( S e. B /\ T e. B /\ ( X ./\ W ) e. B ) ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) = ( ( S .\/ ( X ./\ W ) ) .\/ ( T .\/ ( X ./\ W ) ) ) )
37 15 33 35 21 36 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) = ( ( S .\/ ( X ./\ W ) ) .\/ ( T .\/ ( X ./\ W ) ) ) )
38 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( S .\/ ( X ./\ W ) ) = X )
39 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( T .\/ ( X ./\ W ) ) = X )
40 38 39 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ ( X ./\ W ) ) .\/ ( T .\/ ( X ./\ W ) ) ) = ( X .\/ X ) )
41 1 3 latjidm 
 |-  ( ( K e. Lat /\ X e. B ) -> ( X .\/ X ) = X )
42 15 16 41 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( X .\/ X ) = X )
43 37 40 42 3eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) = X )
44 43 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) )
45 44 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) ) = ( ( S .\/ T ) ./\ ( X ./\ W ) ) )
46 25 31 45 3eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( S .\/ T ) ./\ ( X ./\ W ) ) )
47 simp12r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> -. S .<_ W )
48 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> S =/= T )
49 2 3 4 5 6 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ S =/= T ) ) -> ( ( S .\/ T ) ./\ W ) e. A )
50 8 17 11 47 12 48 49 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ W ) e. A )
51 46 50 eqeltrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) e. A )
52 7 51 eqeltrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> V e. A )