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Metamath Proof Explorer

Theorem cdleme0fN

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

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 )
cdleme0c.3 
|- V = ( ( P .\/ R ) ./\ W )
Assertion cdleme0fN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> V =/= P )

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 cdleme0c.3 
 |-  V = ( ( P .\/ R ) ./\ W )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> K e. HL )
9 8 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> K e. Lat )
10 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> P e. A )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 10 12 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> P e. ( Base ` K ) )
14 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> R e. A )
15 11 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
16 14 15 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> R e. ( Base ` K ) )
17 11 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) e. ( Base ` K ) )
18 9 13 16 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> ( P .\/ R ) e. ( Base ` K ) )
19 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> W e. H )
20 11 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
21 19 20 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> W e. ( Base ` K ) )
22 11 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) .<_ W )
23 9 18 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> ( ( P .\/ R ) ./\ W ) .<_ W )
24 7 23 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> V .<_ W )
25 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> -. P .<_ W )
26 nbrne2 
 |-  ( ( V .<_ W /\ -. P .<_ W ) -> V =/= P )
27 24 25 26 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ R e. A ) ) -> V =/= P )