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Metamath Proof Explorer

Theorem cdleme35h

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one outside of P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 11-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 ) ) )
cdleme35.g 
|- G = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
Assertion cdleme35h 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> R = S )

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 cdleme35.g 
 |-  G = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
9 oveq1 
 |-  ( F = G -> ( F .\/ U ) = ( G .\/ U ) )
10 oveq2 
 |-  ( F = G -> ( Q .\/ F ) = ( Q .\/ G ) )
11 10 oveq1d 
 |-  ( F = G -> ( ( Q .\/ F ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) )
12 11 oveq2d 
 |-  ( F = G -> ( P .\/ ( ( Q .\/ F ) ./\ W ) ) = ( P .\/ ( ( Q .\/ G ) ./\ W ) ) )
13 9 12 oveq12d 
 |-  ( F = G -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) )
14 13 3ad2ant3 
 |-  ( ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) )
15 14 3ad2ant3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) )
16 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
17 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> P =/= Q )
18 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( R e. A /\ -. R .<_ W ) )
19 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> -. R .<_ ( P .\/ Q ) )
20 1 2 3 4 5 6 7 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 )
21 16 17 18 19 20 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = R )
22 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( S e. A /\ -. S .<_ W ) )
23 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> -. S .<_ ( P .\/ Q ) )
24 1 2 3 4 5 6 8 cdleme35g 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) = S )
25 16 17 22 23 24 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) = S )
26 15 21 25 3eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> R = S )