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Metamath Proof Explorer

Theorem cdleme0c

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 12-Jun-2012)

Ref Expression
Hypotheses cdleme0.l 
|- .<_ = ( le ` K )
cdleme0.j 
|- .\/ = ( join ` K )
cdleme0.m 
|- ./\ = ( meet ` K )
cdleme0.a 
|- A = ( Atoms ` K )
cdleme0.h 
|- H = ( LHyp ` K )
cdleme0.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme0c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> U =/= R )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` K )
2 cdleme0.j 
 |-  .\/ = ( join ` K )
3 cdleme0.m 
 |-  ./\ = ( meet ` K )
4 cdleme0.a 
 |-  A = ( Atoms ` K )
5 cdleme0.h 
 |-  H = ( LHyp ` K )
6 cdleme0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> K e. HL )
8 7 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> K e. Lat )
9 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> P e. A )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> P e. ( Base ` K ) )
13 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> Q e. A )
14 10 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> Q e. ( Base ` K ) )
16 10 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
17 8 12 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
18 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> W e. H )
19 10 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> W e. ( Base ` K ) )
21 10 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
22 8 17 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
23 6 22 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> U .<_ W )
24 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> -. R .<_ W )
25 nbrne2 
 |-  ( ( U .<_ W /\ -. R .<_ W ) -> U =/= R )
26 23 24 25 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) ) -> U =/= R )