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Metamath Proof Explorer

Theorem cdleme6

Description: Part of proof of Lemma E in Crawley p. 113. This expresses (r \/ f_s(r)) /\ w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012)

Ref Expression
Hypotheses cdleme4.l 
|- .<_ = ( le ` K )
cdleme4.j 
|- .\/ = ( join ` K )
cdleme4.m 
|- ./\ = ( meet ` K )
cdleme4.a 
|- A = ( Atoms ` K )
cdleme4.h 
|- H = ( LHyp ` K )
cdleme4.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme4.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme4.g 
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
Assertion cdleme6 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = U )

Proof

Step Hyp Ref Expression
1 cdleme4.l 
 |-  .<_ = ( le ` K )
2 cdleme4.j 
 |-  .\/ = ( join ` K )
3 cdleme4.m 
 |-  ./\ = ( meet ` K )
4 cdleme4.a 
 |-  A = ( Atoms ` K )
5 cdleme4.h 
 |-  H = ( LHyp ` K )
6 cdleme4.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme4.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme4.g 
 |-  G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
9 1 2 3 4 5 6 7 8 cdleme5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ G ) = ( P .\/ Q ) )
10 9 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
11 10 6 eqtr4di 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = U )