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Metamath Proof Explorer

Theorem cdleme19f

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, line 3. D , F , N , Y , G , O represent s_2, f(s), f_s(r), t_2, f(t), f_t(r). We prove that if r <_ s \/ t, then f_t(r) = f_t(r). (Contributed by NM, 14-Nov-2012)

Ref Expression
Hypotheses cdleme19.l 
|- .<_ = ( le ` K )
cdleme19.j 
|- .\/ = ( join ` K )
cdleme19.m 
|- ./\ = ( meet ` K )
cdleme19.a 
|- A = ( Atoms ` K )
cdleme19.h 
|- H = ( LHyp ` K )
cdleme19.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme19.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme19.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
cdleme19.o 
|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion cdleme19f 
|- ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> N = O )

Proof

Step Hyp Ref Expression
1 cdleme19.l 
 |-  .<_ = ( le ` K )
2 cdleme19.j 
 |-  .\/ = ( join ` K )
3 cdleme19.m 
 |-  ./\ = ( meet ` K )
4 cdleme19.a 
 |-  A = ( Atoms ` K )
5 cdleme19.h 
 |-  H = ( LHyp ` K )
6 cdleme19.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme19.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme19.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme19.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme19.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme19.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12 cdleme19.o 
 |-  O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
13 1 2 3 4 5 6 7 8 9 10 cdleme19e 
 |-  ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ D ) = ( G .\/ Y ) )
14 13 oveq2d 
 |-  ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ D ) ) = ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) )
15 14 11 12 3eqtr4g 
 |-  ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> N = O )