Metamath Proof Explorer

Theorem cdleme20aN

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 14-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 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 ) )

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 11 oveq1i 
 |-  ( V .\/ D ) = ( ( ( S .\/ T ) ./\ W ) .\/ D )
13 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
14 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
15 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
16 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ W )
17 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
18 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
19 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
20 1 2 3 4 5 9 cdlemeda 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D e. A )
21 13 14 15 16 17 18 19 20 syl223anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> D e. A )
22 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> T e. A )
23 eqid 
 |-  ( Base ` K ) = ( Base ` K )
24 23 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
25 13 15 22 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
26 23 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
27 14 26 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
28 13 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
29 23 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
30 13 17 15 29 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
31 23 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
32 28 30 27 31 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
33 9 32 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> D .<_ W )
34 23 1 2 3 4 atmod4i1 
 |-  ( ( K e. HL /\ ( D e. A /\ ( S .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ D .<_ W ) -> ( ( ( S .\/ T ) ./\ W ) .\/ D ) = ( ( ( S .\/ T ) .\/ D ) ./\ W ) )
35 13 21 25 27 33 34 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ T ) ./\ W ) .\/ D ) = ( ( ( S .\/ T ) .\/ D ) ./\ W ) )
36 1 2 3 4 5 9 cdleme10 
 |-  ( ( ( K e. HL /\ W e. H ) /\ R e. A /\ ( S e. A /\ -. S .<_ W ) ) -> ( S .\/ D ) = ( S .\/ R ) )
37 13 14 17 15 16 36 syl212anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S .\/ D ) = ( S .\/ R ) )
38 37 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ D ) .\/ T ) = ( ( S .\/ R ) .\/ T ) )
39 2 4 hlatj32 
 |-  ( ( K e. HL /\ ( S e. A /\ D e. A /\ T e. A ) ) -> ( ( S .\/ D ) .\/ T ) = ( ( S .\/ T ) .\/ D ) )
40 13 15 21 22 39 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ D ) .\/ T ) = ( ( S .\/ T ) .\/ D ) )
41 38 40 eqtr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ R ) .\/ T ) = ( ( S .\/ T ) .\/ D ) )
42 41 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ R ) .\/ T ) ./\ W ) = ( ( ( S .\/ T ) .\/ D ) ./\ W ) )
43 35 42 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ T ) ./\ W ) .\/ D ) = ( ( ( S .\/ R ) .\/ T ) ./\ W ) )
44 12 43 syl5eq 
 |-  ( ( ( 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 ) )