Metamath Proof Explorer

Theorem cdleme28b

Description: Lemma for cdleme25b . TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013)

Ref Expression
Hypotheses cdleme26.b 
|- B = ( Base ` K )
cdleme26.l 
|- .<_ = ( le ` K )
cdleme26.j 
|- .\/ = ( join ` K )
cdleme26.m 
|- ./\ = ( meet ` K )
cdleme26.a 
|- A = ( Atoms ` K )
cdleme26.h 
|- H = ( LHyp ` K )
cdleme27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme27.f 
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme27.z 
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme27.n 
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
cdleme27.d 
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
cdleme27.c 
|- C = if ( s .<_ ( P .\/ Q ) , D , F )
cdleme27.g 
|- G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme27.o 
|- O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
cdleme27.e 
|- E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
cdleme27.y 
|- Y = if ( t .<_ ( P .\/ Q ) , E , G )
Assertion cdleme28b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b 
 |-  B = ( Base ` K )
2 cdleme26.l 
 |-  .<_ = ( le ` K )
3 cdleme26.j 
 |-  .\/ = ( join ` K )
4 cdleme26.m 
 |-  ./\ = ( meet ` K )
5 cdleme26.a 
 |-  A = ( Atoms ` K )
6 cdleme26.h 
 |-  H = ( LHyp ` K )
7 cdleme27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme27.f 
 |-  F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme27.z 
 |-  Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
10 cdleme27.n 
 |-  N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
11 cdleme27.d 
 |-  D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
12 cdleme27.c 
 |-  C = if ( s .<_ ( P .\/ Q ) , D , F )
13 cdleme27.g 
 |-  G = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
14 cdleme27.o 
 |-  O = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
15 cdleme27.e 
 |-  E = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = O ) )
16 cdleme27.y 
 |-  Y = if ( t .<_ ( P .\/ Q ) , E , G )
17 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> K e. HL )
18 17 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> K e. Lat )
19 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> W e. H )
20 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
21 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
22 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( s e. A /\ -. s .<_ W ) )
23 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> P =/= Q )
24 1 2 3 4 5 6 7 8 9 10 11 12 cdleme27cl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ P =/= Q ) ) -> C e. B )
25 17 19 20 21 22 23 24 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> C e. B )
26 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> X e. B )
27 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
28 19 27 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> W e. B )
29 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
30 18 26 28 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( X ./\ W ) e. B )
31 1 3 latjcl 
 |-  ( ( K e. Lat /\ C e. B /\ ( X ./\ W ) e. B ) -> ( C .\/ ( X ./\ W ) ) e. B )
32 18 25 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) e. B )
33 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( t e. A /\ -. t .<_ W ) )
34 1 2 3 4 5 6 7 13 9 14 15 16 cdleme27cl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ P =/= Q ) ) -> Y e. B )
35 17 19 20 21 33 23 34 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> Y e. B )
36 1 3 latjcl 
 |-  ( ( K e. Lat /\ Y e. B /\ ( X ./\ W ) e. B ) -> ( Y .\/ ( X ./\ W ) ) e. B )
37 18 35 30 36 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Y .\/ ( X ./\ W ) ) e. B )
38 eqid 
 |-  ( ( s .\/ t ) ./\ ( X ./\ W ) ) = ( ( s .\/ t ) ./\ ( X ./\ W ) )
39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 38 cdleme28a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) .<_ ( Y .\/ ( X ./\ W ) ) )
40 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
41 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> s =/= t )
42 41 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> t =/= s )
43 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) )
44 43 ancomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( ( t .\/ ( X ./\ W ) ) = X /\ ( s .\/ ( X ./\ W ) ) = X ) )
45 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( X e. B /\ -. X .<_ W ) )
46 eqid 
 |-  ( ( t .\/ s ) ./\ ( X ./\ W ) ) = ( ( t .\/ s ) ./\ ( X ./\ W ) )
47 1 2 3 4 5 6 7 13 9 14 15 16 8 10 11 12 46 cdleme28a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( t e. A /\ -. t .<_ W ) /\ ( s e. A /\ -. s .<_ W ) ) /\ ( t =/= s /\ ( ( t .\/ ( X ./\ W ) ) = X /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Y .\/ ( X ./\ W ) ) .<_ ( C .\/ ( X ./\ W ) ) )
48 40 20 21 23 33 22 42 44 45 47 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( Y .\/ ( X ./\ W ) ) .<_ ( C .\/ ( X ./\ W ) ) )
49 1 2 18 32 37 39 48 latasymd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( s e. A /\ -. s .<_ W ) /\ ( t e. A /\ -. t .<_ W ) ) /\ ( s =/= t /\ ( ( s .\/ ( X ./\ W ) ) = X /\ ( t .\/ ( X ./\ W ) ) = X ) /\ ( X e. B /\ -. X .<_ W ) ) ) -> ( C .\/ ( X ./\ W ) ) = ( Y .\/ ( X ./\ W ) ) )