Metamath Proof Explorer

Theorem 4atexlemex6

Description: Lemma for 4atexlem7 . (Contributed by NM, 25-Nov-2012)

Ref Expression
Hypotheses 4thatleme.l 
|- .<_ = ( le ` K )
4thatleme.j 
|- .\/ = ( join ` K )
4thatleme.m 
|- ./\ = ( meet ` K )
4thatleme.a 
|- A = ( Atoms ` K )
4thatleme.h 
|- H = ( LHyp ` K )
Assertion 4atexlemex6 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )

Proof

Step Hyp Ref Expression
1 4thatleme.l 
 |-  .<_ = ( le ` K )
2 4thatleme.j 
 |-  .\/ = ( join ` K )
3 4thatleme.m 
 |-  ./\ = ( meet ` K )
4 4thatleme.a 
 |-  A = ( Atoms ` K )
5 4thatleme.h 
 |-  H = ( LHyp ` K )
6 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
7 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
8 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
9 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
10 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
11 1 2 3 4 5 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )
12 7 8 9 10 11 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )
13 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
14 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
15 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
16 1 2 4 atnlej1 
 |-  ( ( K e. HL /\ ( S e. A /\ P e. A /\ Q e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P )
17 6 13 14 9 15 16 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> S =/= P )
18 17 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= S )
19 1 2 3 4 5 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> ( ( P .\/ S ) ./\ W ) e. A )
20 7 8 13 18 19 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ S ) ./\ W ) e. A )
21 2 4 hlsupr2 
 |-  ( ( K e. HL /\ ( ( P .\/ Q ) ./\ W ) e. A /\ ( ( P .\/ S ) ./\ W ) e. A ) -> E. t e. A ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) )
22 6 12 20 21 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> E. t e. A ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) )
23 simp111 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> ( K e. HL /\ W e. H ) )
24 simp112 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> ( P e. A /\ -. P .<_ W ) )
25 simp113 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> ( Q e. A /\ -. Q .<_ W ) )
26 simp12r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> S e. A )
27 simp2ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
28 27 3ad2ant1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> R e. A )
29 simp2lr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. R .<_ W )
30 29 3ad2ant1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> -. R .<_ W )
31 simp131 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> ( P .\/ R ) = ( Q .\/ R ) )
32 28 30 31 3jca 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) )
33 3simpc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> ( t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) )
34 simp132 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> P =/= Q )
35 simp133 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> -. S .<_ ( P .\/ Q ) )
36 biid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
37 eqid 
 |-  ( ( P .\/ Q ) ./\ W ) = ( ( P .\/ Q ) ./\ W )
38 eqid 
 |-  ( ( P .\/ S ) ./\ W ) = ( ( P .\/ S ) ./\ W )
39 eqid 
 |-  ( ( Q .\/ t ) ./\ ( P .\/ S ) ) = ( ( Q .\/ t ) ./\ ( P .\/ S ) )
40 eqid 
 |-  ( ( R .\/ t ) ./\ ( P .\/ S ) ) = ( ( R .\/ t ) ./\ ( P .\/ S ) )
41 36 1 2 3 4 5 37 38 39 40 4atexlemex4 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ ( ( Q .\/ t ) ./\ ( P .\/ S ) ) = S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
42 36 1 2 3 4 5 37 38 39 4atexlemex2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ ( ( Q .\/ t ) ./\ ( P .\/ S ) ) =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
43 41 42 pm2.61dane 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
44 23 24 25 26 32 33 34 35 43 syl332anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) /\ t e. A /\ ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
45 44 rexlimdv3a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( E. t e. A ( ( ( P .\/ Q ) ./\ W ) .\/ t ) = ( ( ( P .\/ S ) ./\ W ) .\/ t ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) )
46 22 45 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ S e. A ) /\ ( ( P .\/ R ) = ( Q .\/ R ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )