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Metamath Proof Explorer

Theorem cdleme21g

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cdleme21.l 
|- .<_ = ( le ` K )
cdleme21.j 
|- .\/ = ( join ` K )
cdleme21.m 
|- ./\ = ( meet ` K )
cdleme21.a 
|- A = ( Atoms ` K )
cdleme21.h 
|- H = ( LHyp ` K )
cdleme21.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme21.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme21g.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme21g.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme21g.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme21g.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
cdleme21g.o 
|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion cdleme21g 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> N = O )

Proof

Step Hyp Ref Expression
1 cdleme21.l 
 |-  .<_ = ( le ` K )
2 cdleme21.j 
 |-  .\/ = ( join ` K )
3 cdleme21.m 
 |-  ./\ = ( meet ` K )
4 cdleme21.a 
 |-  A = ( Atoms ` K )
5 cdleme21.h 
 |-  H = ( LHyp ` K )
6 cdleme21.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme21.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme21g.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme21g.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme21g.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme21g.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12 cdleme21g.o 
 |-  O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
13 eqid 
 |-  ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
14 eqid 
 |-  ( ( R .\/ z ) ./\ W ) = ( ( R .\/ z ) ./\ W )
15 eqid 
 |-  ( ( P .\/ Q ) ./\ ( ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) .\/ ( ( R .\/ z ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) .\/ ( ( R .\/ z ) ./\ W ) ) )
16 1 2 3 4 5 6 7 13 9 14 11 15 8 10 12 cdleme21f 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> N = O )