Metamath Proof Explorer

Theorem cdleme35sn2aw

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one outside of P .\/ Q line case; compare cdleme32sn2awN . TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013)

Ref Expression
Hypotheses cdleme32s.b 
|- B = ( Base ` K )
cdleme32s.l 
|- .<_ = ( le ` K )
cdleme32s.j 
|- .\/ = ( join ` K )
cdleme32s.m 
|- ./\ = ( meet ` K )
cdleme32s.a 
|- A = ( Atoms ` K )
cdleme32s.h 
|- H = ( LHyp ` K )
cdleme32s.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme32s.d 
|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme32s.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , D )
Assertion cdleme35sn2aw 
|- ( ( ( ( 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 ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )

Proof

Step Hyp Ref Expression
1 cdleme32s.b 
 |-  B = ( Base ` K )
2 cdleme32s.l 
 |-  .<_ = ( le ` K )
3 cdleme32s.j 
 |-  .\/ = ( join ` K )
4 cdleme32s.m 
 |-  ./\ = ( meet ` K )
5 cdleme32s.a 
 |-  A = ( Atoms ` K )
6 cdleme32s.h 
 |-  H = ( LHyp ` K )
7 cdleme32s.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme32s.d 
 |-  D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme32s.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , D )
10 eqid 
 |-  ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
11 eqid 
 |-  ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
12 2 3 4 5 6 7 10 11 cdleme35h2 
 |-  ( ( ( ( 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 ) /\ R =/= S ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) =/= ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
13 simp22l 
 |-  ( ( ( ( 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 ) /\ R =/= S ) ) -> R e. A )
14 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 ) /\ R =/= S ) ) -> -. R .<_ ( P .\/ Q ) )
15 8 9 10 cdleme31sn2 
 |-  ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
16 13 14 15 syl2anc 
 |-  ( ( ( ( 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 ) /\ R =/= S ) ) -> [_ R / s ]_ N = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
17 simp23l 
 |-  ( ( ( ( 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 ) /\ R =/= S ) ) -> S e. A )
18 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 ) /\ R =/= S ) ) -> -. S .<_ ( P .\/ Q ) )
19 8 9 11 cdleme31sn2 
 |-  ( ( S e. A /\ -. S .<_ ( P .\/ Q ) ) -> [_ S / s ]_ N = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
20 17 18 19 syl2anc 
 |-  ( ( ( ( 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 ) /\ R =/= S ) ) -> [_ S / s ]_ N = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
21 12 16 20 3netr4d 
 |-  ( ( ( ( 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 ) /\ R =/= S ) ) -> [_ R / s ]_ N =/= [_ S / s ]_ N )