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Metamath Proof Explorer

Theorem cdleme18b

Description: Part of proof of Lemma E in Crawley p. 114, 2nd sentence of 4th paragraph. F , G represent f(s), f_s(q) respectively. We show -. f_s(q) =/= q. (Contributed by NM, 12-Oct-2012)

Ref Expression
Hypotheses cdleme18.l 
|- .<_ = ( le ` K )
cdleme18.j 
|- .\/ = ( join ` K )
cdleme18.m 
|- ./\ = ( meet ` K )
cdleme18.a 
|- A = ( Atoms ` K )
cdleme18.h 
|- H = ( LHyp ` K )
cdleme18.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme18.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme18.g 
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( Q .\/ S ) ./\ W ) ) )
Assertion cdleme18b 
|- ( ( ( 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 ) ) ) -> G =/= Q )

Proof

Step Hyp Ref Expression
1 cdleme18.l 
 |-  .<_ = ( le ` K )
2 cdleme18.j 
 |-  .\/ = ( join ` K )
3 cdleme18.m 
 |-  ./\ = ( meet ` K )
4 cdleme18.a 
 |-  A = ( Atoms ` K )
5 cdleme18.h 
 |-  H = ( LHyp ` K )
6 cdleme18.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme18.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme18.g 
 |-  G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( Q .\/ S ) ./\ W ) ) )
9 eqid 
 |-  Q = Q
10 oveq2 
 |-  ( G = Q -> ( Q .\/ G ) = ( Q .\/ Q ) )
11 simp1l 
 |-  ( ( ( 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 ) ) ) -> K e. HL )
12 simp22l 
 |-  ( ( ( 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 ) ) ) -> Q e. A )
13 2 4 hlatjidm 
 |-  ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q )
14 11 12 13 syl2anc 
 |-  ( ( ( 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 ) ) ) -> ( Q .\/ Q ) = Q )
15 10 14 sylan9eqr 
 |-  ( ( ( ( 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 ) ) ) /\ G = Q ) -> ( Q .\/ G ) = Q )
16 simp1 
 |-  ( ( ( 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 ) ) ) -> ( K e. HL /\ W e. H ) )
17 simp21l 
 |-  ( ( ( 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 ) ) ) -> P e. A )
18 simp22 
 |-  ( ( ( 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 ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
19 simp23 
 |-  ( ( ( 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 ) ) ) -> ( S e. A /\ -. S .<_ W ) )
20 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q .<_ ( P .\/ Q ) )
21 11 17 12 20 syl3anc 
 |-  ( ( ( 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 ) ) ) -> Q .<_ ( P .\/ Q ) )
22 1 2 3 4 5 6 7 8 cdleme5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ Q .<_ ( P .\/ Q ) ) ) -> ( Q .\/ G ) = ( P .\/ Q ) )
23 16 17 12 18 19 21 22 syl132anc 
 |-  ( ( ( 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 ) ) ) -> ( Q .\/ G ) = ( P .\/ Q ) )
24 23 adantr 
 |-  ( ( ( ( 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 ) ) ) /\ G = Q ) -> ( Q .\/ G ) = ( P .\/ Q ) )
25 15 24 eqtr3d 
 |-  ( ( ( ( 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 ) ) ) /\ G = Q ) -> Q = ( P .\/ Q ) )
26 simp3l 
 |-  ( ( ( 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 ) ) ) -> P =/= Q )
27 2 4 2atneat 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> -. ( P .\/ Q ) e. A )
28 11 17 12 26 27 syl13anc 
 |-  ( ( ( 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 ) ) ) -> -. ( P .\/ Q ) e. A )
29 nelne2 
 |-  ( ( Q e. A /\ -. ( P .\/ Q ) e. A ) -> Q =/= ( P .\/ Q ) )
30 29 necomd 
 |-  ( ( Q e. A /\ -. ( P .\/ Q ) e. A ) -> ( P .\/ Q ) =/= Q )
31 12 28 30 syl2anc 
 |-  ( ( ( 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 ) ) ) -> ( P .\/ Q ) =/= Q )
32 31 adantr 
 |-  ( ( ( ( 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 ) ) ) /\ G = Q ) -> ( P .\/ Q ) =/= Q )
33 25 32 eqnetrd 
 |-  ( ( ( ( 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 ) ) ) /\ G = Q ) -> Q =/= Q )
34 33 ex 
 |-  ( ( ( 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 ) ) ) -> ( G = Q -> Q =/= Q ) )
35 34 necon2d 
 |-  ( ( ( 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 ) ) ) -> ( Q = Q -> G =/= Q ) )
36 9 35 mpi 
 |-  ( ( ( 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 ) ) ) -> G =/= Q )