Metamath Proof Explorer


Theorem lhpat2

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 21-Nov-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 )
lhpat2.r
|- R = ( ( P .\/ Q ) ./\ W )
Assertion lhpat2
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> R 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 lhpat2.r
 |-  R = ( ( P .\/ Q ) ./\ W )
7 1 2 3 4 5 lhpat
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )
8 6 7 eqeltrid
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> R e. A )