Metamath Proof Explorer

Theorem cdleme7

Description: Part of proof of Lemma E in Crawley p. 113. G and F represent f_s(r) and f(s) respectively. W is the fiducial co-atom (hyperplane) that they call w. Here and in cdleme7ga above, we show that f_s(r) e. W (top of p. 114), meaning it is an atom and not under w, which in our notation is expressed as G e. A /\ -. G .<_ W . (Note that we do not have a symbol for their W.) Their proof provides no details of our cdleme7aa through cdleme7 , so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012)

Ref Expression
Hypotheses cdleme4.l 
|- .<_ = ( le ` K )
cdleme4.j 
|- .\/ = ( join ` K )
cdleme4.m 
|- ./\ = ( meet ` K )
cdleme4.a 
|- A = ( Atoms ` K )
cdleme4.h 
|- H = ( LHyp ` K )
cdleme4.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme4.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme4.g 
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
Assertion cdleme7 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. G .<_ W )

Proof

Step Hyp Ref Expression
1 cdleme4.l 
 |-  .<_ = ( le ` K )
2 cdleme4.j 
 |-  .\/ = ( join ` K )
3 cdleme4.m 
 |-  ./\ = ( meet ` K )
4 cdleme4.a 
 |-  A = ( Atoms ` K )
5 cdleme4.h 
 |-  H = ( LHyp ` K )
6 cdleme4.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme4.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme4.g 
 |-  G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
9 eqid 
 |-  ( ( R .\/ S ) ./\ W ) = ( ( R .\/ S ) ./\ W )
10 1 2 3 4 5 6 7 8 9 cdleme7d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> G =/= U )
11 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
12 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
13 1 2 3 4 5 6 7 8 cdleme7ga 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> G e. A )
14 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ R e. A /\ G e. A ) -> G .<_ ( R .\/ G ) )
15 11 12 13 14 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> G .<_ ( R .\/ G ) )
16 15 biantrurd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( G .<_ W <-> ( G .<_ ( R .\/ G ) /\ G .<_ W ) ) )
17 11 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
18 eqid 
 |-  ( Base ` K ) = ( Base ` K )
19 18 4 atbase 
 |-  ( G e. A -> G e. ( Base ` K ) )
20 13 19 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> G e. ( Base ` K ) )
21 18 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ G e. A ) -> ( R .\/ G ) e. ( Base ` K ) )
22 11 12 13 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .\/ G ) e. ( Base ` K ) )
23 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
24 18 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
25 23 24 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
26 18 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( G e. ( Base ` K ) /\ ( R .\/ G ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( G .<_ ( R .\/ G ) /\ G .<_ W ) <-> G .<_ ( ( R .\/ G ) ./\ W ) ) )
27 17 20 22 25 26 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( G .<_ ( R .\/ G ) /\ G .<_ W ) <-> G .<_ ( ( R .\/ G ) ./\ W ) ) )
28 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
29 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
30 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
31 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
32 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
33 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
34 1 2 3 4 5 6 7 8 cdleme6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = U )
35 28 29 30 31 32 33 34 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = U )
36 35 breq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( G .<_ ( ( R .\/ G ) ./\ W ) <-> G .<_ U ) )
37 27 36 bitrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( G .<_ ( R .\/ G ) /\ G .<_ W ) <-> G .<_ U ) )
38 hlatl 
 |-  ( K e. HL -> K e. AtLat )
39 11 38 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. AtLat )
40 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
41 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
42 1 2 3 4 5 6 lhpat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
43 28 40 30 41 42 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U e. A )
44 1 4 atcmp 
 |-  ( ( K e. AtLat /\ G e. A /\ U e. A ) -> ( G .<_ U <-> G = U ) )
45 39 13 43 44 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 /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( G .<_ U <-> G = U ) )
46 16 37 45 3bitrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( G .<_ W <-> G = U ) )
47 46 necon3bbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( -. G .<_ W <-> G =/= U ) )
48 10 47 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. G .<_ W )