Metamath Proof Explorer


Theorem cdleme0gN

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 cdleme0gN
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ R e. A ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> V =/= Q )

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 1 2 3 4 5 7 cdleme0c
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ R e. A ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> V =/= Q )