Metamath Proof Explorer

Theorem cdleme11h

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 14-Jun-2012)

Ref Expression
Hypotheses cdleme11.l 
|- .<_ = ( le ` K )
cdleme11.j 
|- .\/ = ( join ` K )
cdleme11.m 
|- ./\ = ( meet ` K )
cdleme11.a 
|- A = ( Atoms ` K )
cdleme11.h 
|- H = ( LHyp ` K )
cdleme11.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme11.c 
|- C = ( ( P .\/ S ) ./\ W )
cdleme11.d 
|- D = ( ( P .\/ T ) ./\ W )
cdleme11.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
Assertion cdleme11h 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= Q )

Proof

Step Hyp Ref Expression
1 cdleme11.l 
 |-  .<_ = ( le ` K )
2 cdleme11.j 
 |-  .\/ = ( join ` K )
3 cdleme11.m 
 |-  ./\ = ( meet ` K )
4 cdleme11.a 
 |-  A = ( Atoms ` K )
5 cdleme11.h 
 |-  H = ( LHyp ` K )
6 cdleme11.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme11.c 
 |-  C = ( ( P .\/ S ) ./\ W )
8 cdleme11.d 
 |-  D = ( ( P .\/ T ) ./\ W )
9 cdleme11.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
10 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
12 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
13 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
14 simp22r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. Q .<_ W )
15 1 2 3 4 5 7 cdleme0c 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ S e. A ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> C =/= Q )
16 10 11 12 13 14 15 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> C =/= Q )
17 16 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> Q =/= C )
18 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
19 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
20 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
21 1 2 3 4 cdleme00a 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P )
22 18 11 13 12 20 21 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S =/= P )
23 22 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= S )
24 1 2 3 4 5 7 cdleme9a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> C e. A )
25 10 19 12 23 24 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> C e. A )
26 2 4 lnnat 
 |-  ( ( K e. HL /\ Q e. A /\ C e. A ) -> ( Q =/= C <-> -. ( Q .\/ C ) e. A ) )
27 18 13 25 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q =/= C <-> -. ( Q .\/ C ) e. A ) )
28 17 27 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. ( Q .\/ C ) e. A )
29 2 4 hlatjidm 
 |-  ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q )
30 18 13 29 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q .\/ Q ) = Q )
31 30 13 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q .\/ Q ) e. A )
32 oveq2 
 |-  ( F = Q -> ( Q .\/ F ) = ( Q .\/ Q ) )
33 32 eleq1d 
 |-  ( F = Q -> ( ( Q .\/ F ) e. A <-> ( Q .\/ Q ) e. A ) )
34 31 33 syl5ibrcom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F = Q -> ( Q .\/ F ) e. A ) )
35 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
36 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
37 1 2 3 4 5 6 7 6 9 cdleme11g 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ P =/= Q ) -> ( Q .\/ F ) = ( Q .\/ C ) )
38 10 11 35 12 36 37 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q .\/ F ) = ( Q .\/ C ) )
39 38 eleq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( Q .\/ F ) e. A <-> ( Q .\/ C ) e. A ) )
40 34 39 sylibd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F = Q -> ( Q .\/ C ) e. A ) )
41 40 necon3bd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( -. ( Q .\/ C ) e. A -> F =/= Q ) )
42 28 41 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= Q )