Metamath Proof Explorer

Theorem cdleme19a

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, 1st line. D represents s_2. In their notation, we prove that if r <_ s \/ t, then s_2=(s \/ t) /\ w. (Contributed by NM, 13-Nov-2012)

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 )
Assertion cdleme19a 
|- ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> D = ( ( S .\/ T ) ./\ W ) )

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 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 hllat 
 |-  ( K e. HL -> K e. Lat )
13 12 3ad2ant1 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> K e. Lat )
14 simp1 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> K e. HL )
15 simp21 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> R e. A )
16 simp22 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> S e. A )
17 11 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
18 14 15 16 17 syl3anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
19 simp23 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> T e. A )
20 11 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
21 14 16 19 20 syl3anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
22 simp33 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> R .<_ ( S .\/ T ) )
23 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> S .<_ ( S .\/ T ) )
24 14 16 19 23 syl3anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> S .<_ ( S .\/ T ) )
25 11 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
26 15 25 syl 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> R e. ( Base ` K ) )
27 11 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
28 16 27 syl 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> S e. ( Base ` K ) )
29 11 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ S e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) ) -> ( ( R .<_ ( S .\/ T ) /\ S .<_ ( S .\/ T ) ) <-> ( R .\/ S ) .<_ ( S .\/ T ) ) )
30 13 26 28 21 29 syl13anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( ( R .<_ ( S .\/ T ) /\ S .<_ ( S .\/ T ) ) <-> ( R .\/ S ) .<_ ( S .\/ T ) ) )
31 22 24 30 mpbi2and 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( R .\/ S ) .<_ ( S .\/ T ) )
32 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> S .<_ ( R .\/ S ) )
33 14 15 16 32 syl3anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> S .<_ ( R .\/ S ) )
34 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
35 34 3ad2ant1 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> K e. CvLat )
36 simp31 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> R .<_ ( P .\/ Q ) )
37 simp32 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) )
38 nbrne2 
 |-  ( ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) -> R =/= S )
39 36 37 38 syl2anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> R =/= S )
40 1 2 4 cvlatexch1 
 |-  ( ( K e. CvLat /\ ( R e. A /\ T e. A /\ S e. A ) /\ R =/= S ) -> ( R .<_ ( S .\/ T ) -> T .<_ ( S .\/ R ) ) )
41 35 15 19 16 39 40 syl131anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( R .<_ ( S .\/ T ) -> T .<_ ( S .\/ R ) ) )
42 22 41 mpd 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> T .<_ ( S .\/ R ) )
43 2 4 hlatjcom 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) = ( S .\/ R ) )
44 14 15 16 43 syl3anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( R .\/ S ) = ( S .\/ R ) )
45 42 44 breqtrrd 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> T .<_ ( R .\/ S ) )
46 11 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
47 19 46 syl 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> T e. ( Base ` K ) )
48 11 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ T e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) ) ) -> ( ( S .<_ ( R .\/ S ) /\ T .<_ ( R .\/ S ) ) <-> ( S .\/ T ) .<_ ( R .\/ S ) ) )
49 13 28 47 18 48 syl13anc 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( ( S .<_ ( R .\/ S ) /\ T .<_ ( R .\/ S ) ) <-> ( S .\/ T ) .<_ ( R .\/ S ) ) )
50 33 45 49 mpbi2and 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) .<_ ( R .\/ S ) )
51 11 1 13 18 21 31 50 latasymd 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( R .\/ S ) = ( S .\/ T ) )
52 51 oveq1d 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> ( ( R .\/ S ) ./\ W ) = ( ( S .\/ T ) ./\ W ) )
53 9 52 syl5eq 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) -> D = ( ( S .\/ T ) ./\ W ) )