Metamath Proof Explorer

Theorem cdleme35g

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

Ref Expression
Hypotheses cdleme35.l 
|- .<_ = ( le ` K )
cdleme35.j 
|- .\/ = ( join ` K )
cdleme35.m 
|- ./\ = ( meet ` K )
cdleme35.a 
|- A = ( Atoms ` K )
cdleme35.h 
|- H = ( LHyp ` K )
cdleme35.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme35.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdleme35g 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = R )

Proof

Step Hyp Ref Expression
1 cdleme35.l 
 |-  .<_ = ( le ` K )
2 cdleme35.j 
 |-  .\/ = ( join ` K )
3 cdleme35.m 
 |-  ./\ = ( meet ` K )
4 cdleme35.a 
 |-  A = ( Atoms ` K )
5 cdleme35.h 
 |-  H = ( LHyp ` K )
6 cdleme35.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme35.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 1 2 3 4 5 6 7 cdleme35a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F .\/ U ) = ( R .\/ U ) )
9 1 2 3 4 5 6 7 cdleme35e 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ ( ( Q .\/ F ) ./\ W ) ) = ( P .\/ R ) )
10 8 9 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( P .\/ R ) ) )
11 1 2 3 4 5 6 7 cdleme35f 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( R .\/ U ) ./\ ( P .\/ R ) ) = R )
12 10 11 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = R )