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Metamath Proof Explorer

Theorem lhpat

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 23-May-2012)

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

Proof

Step Hyp Ref Expression
1 lhpat.l 
 |-  .<_ = ( le ` K )
2 lhpat.j 
 |-  .\/ = ( join ` K )
3 lhpat.m 
 |-  ./\ = ( meet ` K )
4 lhpat.a 
 |-  A = ( Atoms ` K )
5 lhpat.h 
 |-  H = ( LHyp ` K )
6 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> K e. HL )
7 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> P e. A )
8 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> Q e. A )
9 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> W e. H )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> W e. ( Base ` K ) )
13 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> P =/= Q )
14 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
15 eqid 
 |-  (
16 14 15 5 lhp1cvr 
 |-  ( ( K e. HL /\ W e. H ) -> W (
17 16 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> W (
18 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> -. P .<_ W )
19 10 1 2 3 14 15 4 1cvrat 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ W e. ( Base ` K ) ) /\ ( P =/= Q /\ W (  ( ( P .\/ Q ) ./\ W ) e. A )
20 6 7 8 12 13 17 18 19 syl133anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )