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Metamath Proof Explorer

Theorem cdleme23a

Description: Part of proof of Lemma E in Crawley p. 113. (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 cdleme23a 
|- ( ( ( ( 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 .<_ W )

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 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 )
10 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 )
11 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 )
12 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. B )
13 8 10 11 12 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 )
14 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 )
15 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 )
16 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
17 15 16 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 )
18 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
19 9 14 17 18 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 )
20 1 4 latmcl 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) e. B )
21 9 13 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 ) ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) e. B )
22 1 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) .<_ ( X ./\ W ) )
23 9 13 19 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 ) ) .<_ ( X ./\ W ) )
24 1 2 4 latmle2 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W )
25 9 14 17 24 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 ) .<_ W )
26 1 2 9 21 19 17 23 25 lattrd 
 |-  ( ( ( ( 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 )
27 7 26 eqbrtrid 
 |-  ( ( ( ( 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 .<_ W )