Metamath Proof Explorer


Theorem cdleme3fa

Description: Part of proof of Lemma E in Crawley p. 113. See cdleme3 . (Contributed by NM, 6-Oct-2012)

Ref Expression
Hypotheses cdleme1.l
|- .<_ = ( le ` K )
cdleme1.j
|- .\/ = ( join ` K )
cdleme1.m
|- ./\ = ( meet ` K )
cdleme1.a
|- A = ( Atoms ` K )
cdleme1.h
|- H = ( LHyp ` K )
cdleme1.u
|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdleme3fa
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> F e. A )

Proof

Step Hyp Ref Expression
1 cdleme1.l
 |-  .<_ = ( le ` K )
2 cdleme1.j
 |-  .\/ = ( join ` K )
3 cdleme1.m
 |-  ./\ = ( meet ` K )
4 cdleme1.a
 |-  A = ( Atoms ` K )
5 cdleme1.h
 |-  H = ( LHyp ` K )
6 cdleme1.u
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme1.f
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 eqid
 |-  ( ( P .\/ R ) ./\ W ) = ( ( P .\/ R ) ./\ W )
9 1 2 3 4 5 6 7 8 cdleme3h
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> F e. A )