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Metamath Proof Explorer

Theorem cdleme17d1

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G represent f(s), f_s(p) respectively. We show, in their notation, f_s(p)=q. (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses cdleme17.l 
|- .<_ = ( le ` K )
cdleme17.j 
|- .\/ = ( join ` K )
cdleme17.m 
|- ./\ = ( meet ` K )
cdleme17.a 
|- A = ( Atoms ` K )
cdleme17.h 
|- H = ( LHyp ` K )
cdleme17.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme17.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme17.g 
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
Assertion cdleme17d1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = Q )

Proof

Step Hyp Ref Expression
1 cdleme17.l 
 |-  .<_ = ( le ` K )
2 cdleme17.j 
 |-  .\/ = ( join ` K )
3 cdleme17.m 
 |-  ./\ = ( meet ` K )
4 cdleme17.a 
 |-  A = ( Atoms ` K )
5 cdleme17.h 
 |-  H = ( LHyp ` K )
6 cdleme17.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme17.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme17.g 
 |-  G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
9 eqid 
 |-  ( ( P .\/ S ) ./\ W ) = ( ( P .\/ S ) ./\ W )
10 1 2 3 4 5 6 7 8 9 cdleme17a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = ( ( P .\/ Q ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
11 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> K e. HL )
12 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> W e. H )
13 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P e. A )
14 simp21r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. P .<_ W )
15 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> Q e. A )
16 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S e. A )
17 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. S .<_ ( P .\/ Q ) )
18 1 2 3 4 5 6 7 8 9 cdleme17c 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = Q )
19 11 12 13 14 15 16 17 18 syl223anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = Q )
20 10 19 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = Q )