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Metamath Proof Explorer

Theorem cdleme19c

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, 1st line. D , F represent s_2, f(s). We prove f(s) =/= s_2. (Contributed by NM, 13-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 )
Assertion cdleme19c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= D )

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 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
12 11 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
13 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
14 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
17 11 13 14 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
18 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
19 15 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
21 15 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
22 12 17 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
23 9 22 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> D .<_ W )
24 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
25 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
26 24 25 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) )
27 1 2 3 4 5 6 7 cdleme3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W )
28 26 27 syld3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. F .<_ W )
29 nbrne2 
 |-  ( ( D .<_ W /\ -. F .<_ W ) -> D =/= F )
30 23 28 29 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> D =/= F )
31 30 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= D )