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Metamath Proof Explorer

Theorem cdleme20k

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, antepenultimate line. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 20-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 )
cdleme20.v 
|- V = ( ( S .\/ T ) ./\ W )
Assertion cdleme20k 
|- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ D ) =/= ( P .\/ Q ) )

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 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
13 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> P e. A )
14 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> Q e. A )
15 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
16 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
17 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
18 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
19 1 2 3 4 5 9 cdlemednpq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. D .<_ ( P .\/ Q ) )
20 12 13 14 15 16 17 18 19 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. D .<_ ( P .\/ Q ) )
21 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
22 21 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
23 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
24 simp2ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
25 eqid 
 |-  ( Base ` K ) = ( Base ` K )
26 1 2 3 4 5 6 7 25 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
27 21 23 13 14 24 26 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> F e. ( Base ` K ) )
28 simp2rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
29 1 2 3 4 5 9 25 cdlemedb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ S e. A ) ) -> D e. ( Base ` K ) )
30 21 23 28 24 29 syl22anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> D e. ( Base ` K ) )
31 25 1 2 latlej2 
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ D e. ( Base ` K ) ) -> D .<_ ( F .\/ D ) )
32 22 27 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> D .<_ ( F .\/ D ) )
33 breq2 
 |-  ( ( F .\/ D ) = ( P .\/ Q ) -> ( D .<_ ( F .\/ D ) <-> D .<_ ( P .\/ Q ) ) )
34 32 33 syl5ibcom 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( F .\/ D ) = ( P .\/ Q ) -> D .<_ ( P .\/ Q ) ) )
35 34 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( -. D .<_ ( P .\/ Q ) -> ( F .\/ D ) =/= ( P .\/ Q ) ) )
36 20 35 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ D ) =/= ( P .\/ Q ) )