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Metamath Proof Explorer

Theorem cdleme16g

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, Eq. (1). F and G represent f(s) and f(t) respectively. We show, in their notation, (s \/ t) /\ w=(f(s) \/ f(t)) /\ w. (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses cdleme12.l 
|- .<_ = ( le ` K )
cdleme12.j 
|- .\/ = ( join ` K )
cdleme12.m 
|- ./\ = ( meet ` K )
cdleme12.a 
|- A = ( Atoms ` K )
cdleme12.h 
|- H = ( LHyp ` K )
cdleme12.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme12.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme12.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
Assertion cdleme16g 
|- ( ( ( ( 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 =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( F .\/ G ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l 
 |-  .<_ = ( le ` K )
2 cdleme12.j 
 |-  .\/ = ( join ` K )
3 cdleme12.m 
 |-  ./\ = ( meet ` K )
4 cdleme12.a 
 |-  A = ( Atoms ` K )
5 cdleme12.h 
 |-  H = ( LHyp ` K )
6 cdleme12.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme12.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme12.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 1 2 3 4 5 6 7 8 cdleme16e 
 |-  ( ( ( ( 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 =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) = ( ( S .\/ T ) ./\ W ) )
10 1 2 3 4 5 6 7 8 cdleme16f 
 |-  ( ( ( ( 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 =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ ( F .\/ G ) ) = ( ( F .\/ G ) ./\ W ) )
11 9 10 eqtr3d 
 |-  ( ( ( ( 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 =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ -. U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( F .\/ G ) ./\ W ) )