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Metamath Proof Explorer

Theorem cdleme11fN

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 14-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme11.l 
|- .<_ = ( le ` K )
cdleme11.j 
|- .\/ = ( join ` K )
cdleme11.m 
|- ./\ = ( meet ` K )
cdleme11.a 
|- A = ( Atoms ` K )
cdleme11.h 
|- H = ( LHyp ` K )
cdleme11.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme11.c 
|- C = ( ( P .\/ S ) ./\ W )
cdleme11.d 
|- D = ( ( P .\/ T ) ./\ W )
cdleme11.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
Assertion cdleme11fN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= C )

Proof

Step Hyp Ref Expression
1 cdleme11.l 
 |-  .<_ = ( le ` K )
2 cdleme11.j 
 |-  .\/ = ( join ` K )
3 cdleme11.m 
 |-  ./\ = ( meet ` K )
4 cdleme11.a 
 |-  A = ( Atoms ` K )
5 cdleme11.h 
 |-  H = ( LHyp ` K )
6 cdleme11.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme11.c 
 |-  C = ( ( P .\/ S ) ./\ W )
8 cdleme11.d 
 |-  D = ( ( P .\/ T ) ./\ W )
9 cdleme11.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
10 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
12 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
15 12 14 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) )
16 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
17 13 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
18 16 17 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. ( Base ` K ) )
19 13 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( P .\/ S ) e. ( Base ` K ) )
20 11 15 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ S ) e. ( Base ` K ) )
21 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
22 13 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
24 13 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
25 11 20 23 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
26 7 25 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> C .<_ W )
27 1 2 3 4 5 6 9 cdleme3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W )
28 nbrne2 
 |-  ( ( C .<_ W /\ -. F .<_ W ) -> C =/= F )
29 28 necomd 
 |-  ( ( C .<_ W /\ -. F .<_ W ) -> F =/= C )
30 26 27 29 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= C )