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Metamath Proof Explorer

Theorem cdleme23c

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 cdleme23c 
|- ( ( ( ( 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 .\/ V ) )

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 1 5 atbase 
 |-  ( S e. A -> S e. B )
12 10 11 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 )
13 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 )
14 1 5 atbase 
 |-  ( T e. A -> T e. B )
15 13 14 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 )
16 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ S e. B /\ T e. B ) -> S .<_ ( S .\/ T ) )
17 9 12 15 16 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 .<_ ( S .\/ T ) )
18 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 )
19 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 )
20 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
21 19 20 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 )
22 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
23 9 18 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 ) ) -> ( X ./\ W ) e. B )
24 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ S e. B /\ ( X ./\ W ) e. B ) -> S .<_ ( S .\/ ( X ./\ W ) ) )
25 9 12 23 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 ) ) -> S .<_ ( S .\/ ( X ./\ W ) ) )
26 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 )
27 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 )
28 26 27 eqtr4d 
 |-  ( ( ( ( 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 ) ) )
29 25 28 breqtrd 
 |-  ( ( ( ( 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 ) ) )
30 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. B )
31 8 10 13 30 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 )
32 1 3 latjcl 
 |-  ( ( K e. Lat /\ T e. B /\ ( X ./\ W ) e. B ) -> ( T .\/ ( X ./\ W ) ) e. B )
33 9 15 23 32 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 ) ) -> ( T .\/ ( X ./\ W ) ) e. B )
34 1 2 4 latlem12 
 |-  ( ( K e. Lat /\ ( S e. B /\ ( S .\/ T ) e. B /\ ( T .\/ ( X ./\ W ) ) e. B ) ) -> ( ( S .<_ ( S .\/ T ) /\ S .<_ ( T .\/ ( X ./\ W ) ) ) <-> S .<_ ( ( S .\/ T ) ./\ ( T .\/ ( X ./\ W ) ) ) ) )
35 9 12 31 33 34 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 .<_ ( S .\/ T ) /\ S .<_ ( T .\/ ( X ./\ W ) ) ) <-> S .<_ ( ( S .\/ T ) ./\ ( T .\/ ( X ./\ W ) ) ) ) )
36 17 29 35 mpbi2and 
 |-  ( ( ( ( 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 .<_ ( ( S .\/ T ) ./\ ( T .\/ ( X ./\ W ) ) ) )
37 7 oveq2i 
 |-  ( T .\/ V ) = ( T .\/ ( ( S .\/ T ) ./\ ( X ./\ W ) ) )
38 1 2 3 latlej2 
 |-  ( ( K e. Lat /\ S e. B /\ T e. B ) -> T .<_ ( S .\/ T ) )
39 9 12 15 38 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 ) ) -> T .<_ ( S .\/ T ) )
40 1 2 3 4 5 atmod3i1 
 |-  ( ( K e. HL /\ ( T e. A /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) /\ T .<_ ( S .\/ T ) ) -> ( T .\/ ( ( S .\/ T ) ./\ ( X ./\ W ) ) ) = ( ( S .\/ T ) ./\ ( T .\/ ( X ./\ W ) ) ) )
41 8 13 31 23 39 40 syl131anc 
 |-  ( ( ( ( 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 .\/ ( ( S .\/ T ) ./\ ( X ./\ W ) ) ) = ( ( S .\/ T ) ./\ ( T .\/ ( X ./\ W ) ) ) )
42 37 41 eqtrid 
 |-  ( ( ( ( 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 .\/ V ) = ( ( S .\/ T ) ./\ ( T .\/ ( X ./\ W ) ) ) )
43 36 42 breqtrrd 
 |-  ( ( ( ( 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 .\/ V ) )