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Metamath Proof Explorer

Theorem cdleme20bN

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, second line. D , F , Y , G represent s_2, f(s), t_2, f(t). We show v \/ s_2 = v \/ t_2. (Contributed by NM, 15-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme19.l 
|- .<_ = ( le ` K )
cdleme19.j 
|- .\/ = ( join ` K )
cdleme19.m 
|- ./\ = ( meet ` K )
cdleme19.a 
|- A = ( Atoms ` K )
cdleme19.h 
|- H = ( LHyp ` K )
cdleme19.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme19.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme20.v 
|- V = ( ( S .\/ T ) ./\ W )
Assertion cdleme20bN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( V .\/ D ) = ( V .\/ Y ) )

Proof

Step Hyp Ref Expression
1 cdleme19.l 
 |-  .<_ = ( le ` K )
2 cdleme19.j 
 |-  .\/ = ( join ` K )
3 cdleme19.m 
 |-  ./\ = ( meet ` K )
4 cdleme19.a 
 |-  A = ( Atoms ` K )
5 cdleme19.h 
 |-  H = ( LHyp ` K )
6 cdleme19.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme19.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme19.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme19.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme19.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme20.v 
 |-  V = ( ( S .\/ T ) ./\ W )
12 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
13 12 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
14 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
17 14 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. ( Base ` K ) )
18 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
19 15 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. ( Base ` K ) )
21 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> T e. A )
22 15 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> T e. ( Base ` K ) )
24 15 2 latj31 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ R e. ( Base ` K ) /\ T e. ( Base ` K ) ) ) -> ( ( S .\/ R ) .\/ T ) = ( ( T .\/ R ) .\/ S ) )
25 13 17 20 23 24 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ R ) .\/ T ) = ( ( T .\/ R ) .\/ S ) )
26 25 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ R ) .\/ T ) ./\ W ) = ( ( ( T .\/ R ) .\/ S ) ./\ W ) )
27 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
28 simp22r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ W )
29 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
30 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
31 1 2 3 4 5 6 7 8 9 10 11 cdleme20aN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( V .\/ D ) = ( ( ( S .\/ R ) .\/ T ) ./\ W ) )
32 12 27 18 14 28 21 29 30 31 syl233anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( V .\/ D ) = ( ( ( S .\/ R ) .\/ T ) ./\ W ) )
33 2 4 hlatjcom 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) = ( T .\/ S ) )
34 12 14 21 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S .\/ T ) = ( T .\/ S ) )
35 34 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( T .\/ S ) ./\ W ) )
36 11 35 eqtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> V = ( ( T .\/ S ) ./\ W ) )
37 36 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( V .\/ Y ) = ( ( ( T .\/ S ) ./\ W ) .\/ Y ) )
38 simp23r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. T .<_ W )
39 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. T .<_ ( P .\/ Q ) )
40 eqid 
 |-  ( ( T .\/ S ) ./\ W ) = ( ( T .\/ S ) ./\ W )
41 1 2 3 4 5 6 8 7 10 9 40 cdleme20aN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ T e. A /\ -. T .<_ W ) /\ ( S e. A /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( T .\/ S ) ./\ W ) .\/ Y ) = ( ( ( T .\/ R ) .\/ S ) ./\ W ) )
42 12 27 18 21 38 14 39 30 41 syl233anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( T .\/ S ) ./\ W ) .\/ Y ) = ( ( ( T .\/ R ) .\/ S ) ./\ W ) )
43 37 42 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( V .\/ Y ) = ( ( ( T .\/ R ) .\/ S ) ./\ W ) )
44 26 32 43 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( V .\/ D ) = ( V .\/ Y ) )