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Metamath Proof Explorer

Theorem cdleme42d

Description: Part of proof of Lemma E in Crawley p. 113. Match ( s .\/ ( x ./\ W ) ) = x . (Contributed by NM, 6-Mar-2013)

Ref Expression
Hypotheses cdleme42.b 
|- B = ( Base ` K )
cdleme42.l 
|- .<_ = ( le ` K )
cdleme42.j 
|- .\/ = ( join ` K )
cdleme42.m 
|- ./\ = ( meet ` K )
cdleme42.a 
|- A = ( Atoms ` K )
cdleme42.h 
|- H = ( LHyp ` K )
cdleme42.v 
|- V = ( ( R .\/ S ) ./\ W )
Assertion cdleme42d 
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ ( ( R .\/ V ) ./\ W ) ) = ( R .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdleme42.b 
 |-  B = ( Base ` K )
2 cdleme42.l 
 |-  .<_ = ( le ` K )
3 cdleme42.j 
 |-  .\/ = ( join ` K )
4 cdleme42.m 
 |-  ./\ = ( meet ` K )
5 cdleme42.a 
 |-  A = ( Atoms ` K )
6 cdleme42.h 
 |-  H = ( LHyp ` K )
7 cdleme42.v 
 |-  V = ( ( R .\/ S ) ./\ W )
8 7 oveq2i 
 |-  ( R .\/ V ) = ( R .\/ ( ( R .\/ S ) ./\ W ) )
9 1 2 3 4 5 6 7 cdleme42a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) = ( R .\/ V ) )
10 9 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( ( R .\/ S ) ./\ W ) = ( ( R .\/ V ) ./\ W ) )
11 10 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ ( ( R .\/ S ) ./\ W ) ) = ( R .\/ ( ( R .\/ V ) ./\ W ) ) )
12 8 11 eqtr2id 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ ( ( R .\/ V ) ./\ W ) ) = ( R .\/ V ) )