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Metamath Proof Explorer

Theorem 4atexlemnclw

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses 4thatlem.ph 
|- ( ph <-> ( ( ( 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 /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
4thatlem0.l 
|- .<_ = ( le ` K )
4thatlem0.j 
|- .\/ = ( join ` K )
4thatlem0.m 
|- ./\ = ( meet ` K )
4thatlem0.a 
|- A = ( Atoms ` K )
4thatlem0.h 
|- H = ( LHyp ` K )
4thatlem0.u 
|- U = ( ( P .\/ Q ) ./\ W )
4thatlem0.v 
|- V = ( ( P .\/ S ) ./\ W )
4thatlem0.c 
|- C = ( ( Q .\/ T ) ./\ ( P .\/ S ) )
Assertion 4atexlemnclw 
|- ( ph -> -. C .<_ W )

Proof

Step Hyp Ref Expression
1 4thatlem.ph 
 |-  ( ph <-> ( ( ( 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 /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
2 4thatlem0.l 
 |-  .<_ = ( le ` K )
3 4thatlem0.j 
 |-  .\/ = ( join ` K )
4 4thatlem0.m 
 |-  ./\ = ( meet ` K )
5 4thatlem0.a 
 |-  A = ( Atoms ` K )
6 4thatlem0.h 
 |-  H = ( LHyp ` K )
7 4thatlem0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 4thatlem0.v 
 |-  V = ( ( P .\/ S ) ./\ W )
9 4thatlem0.c 
 |-  C = ( ( Q .\/ T ) ./\ ( P .\/ S ) )
10 1 4atexlemkl 
 |-  ( ph -> K e. Lat )
11 1 3 5 4atexlemqtb 
 |-  ( ph -> ( Q .\/ T ) e. ( Base ` K ) )
12 1 3 5 4atexlempsb 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
15 10 11 12 14 syl3anc 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
16 9 15 eqbrtrid 
 |-  ( ph -> C .<_ ( Q .\/ T ) )
17 simp13r 
 |-  ( ( ( ( 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 /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. Q .<_ W )
18 1 17 sylbi 
 |-  ( ph -> -. Q .<_ W )
19 1 4atexlemkc 
 |-  ( ph -> K e. CvLat )
20 1 2 3 4 5 6 7 8 4atexlemv 
 |-  ( ph -> V e. A )
21 1 4atexlemq 
 |-  ( ph -> Q e. A )
22 1 4atexlemt 
 |-  ( ph -> T e. A )
23 1 2 3 4 5 6 7 4atexlemu 
 |-  ( ph -> U e. A )
24 1 2 3 4 5 6 7 8 4atexlemunv 
 |-  ( ph -> U =/= V )
25 1 4atexlemutvt 
 |-  ( ph -> ( U .\/ T ) = ( V .\/ T ) )
26 5 3 cvlsupr6 
 |-  ( ( K e. CvLat /\ ( U e. A /\ V e. A /\ T e. A ) /\ ( U =/= V /\ ( U .\/ T ) = ( V .\/ T ) ) ) -> T =/= V )
27 26 necomd 
 |-  ( ( K e. CvLat /\ ( U e. A /\ V e. A /\ T e. A ) /\ ( U =/= V /\ ( U .\/ T ) = ( V .\/ T ) ) ) -> V =/= T )
28 19 23 20 22 24 25 27 syl132anc 
 |-  ( ph -> V =/= T )
29 2 3 5 cvlatexch2 
 |-  ( ( K e. CvLat /\ ( V e. A /\ Q e. A /\ T e. A ) /\ V =/= T ) -> ( V .<_ ( Q .\/ T ) -> Q .<_ ( V .\/ T ) ) )
30 19 20 21 22 28 29 syl131anc 
 |-  ( ph -> ( V .<_ ( Q .\/ T ) -> Q .<_ ( V .\/ T ) ) )
31 1 6 4atexlemwb 
 |-  ( ph -> W e. ( Base ` K ) )
32 13 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
33 10 12 31 32 syl3anc 
 |-  ( ph -> ( ( P .\/ S ) ./\ W ) .<_ W )
34 8 33 eqbrtrid 
 |-  ( ph -> V .<_ W )
35 1 2 3 4 5 6 7 8 4atexlemtlw 
 |-  ( ph -> T .<_ W )
36 13 5 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
37 20 36 syl 
 |-  ( ph -> V e. ( Base ` K ) )
38 13 5 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
39 22 38 syl 
 |-  ( ph -> T e. ( Base ` K ) )
40 13 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( V e. ( Base ` K ) /\ T e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( V .<_ W /\ T .<_ W ) <-> ( V .\/ T ) .<_ W ) )
41 10 37 39 31 40 syl13anc 
 |-  ( ph -> ( ( V .<_ W /\ T .<_ W ) <-> ( V .\/ T ) .<_ W ) )
42 34 35 41 mpbi2and 
 |-  ( ph -> ( V .\/ T ) .<_ W )
43 13 5 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
44 21 43 syl 
 |-  ( ph -> Q e. ( Base ` K ) )
45 1 4atexlemk 
 |-  ( ph -> K e. HL )
46 13 3 5 hlatjcl 
 |-  ( ( K e. HL /\ V e. A /\ T e. A ) -> ( V .\/ T ) e. ( Base ` K ) )
47 45 20 22 46 syl3anc 
 |-  ( ph -> ( V .\/ T ) e. ( Base ` K ) )
48 13 2 lattr 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ ( V .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( Q .<_ ( V .\/ T ) /\ ( V .\/ T ) .<_ W ) -> Q .<_ W ) )
49 10 44 47 31 48 syl13anc 
 |-  ( ph -> ( ( Q .<_ ( V .\/ T ) /\ ( V .\/ T ) .<_ W ) -> Q .<_ W ) )
50 42 49 mpan2d 
 |-  ( ph -> ( Q .<_ ( V .\/ T ) -> Q .<_ W ) )
51 30 50 syld 
 |-  ( ph -> ( V .<_ ( Q .\/ T ) -> Q .<_ W ) )
52 18 51 mtod 
 |-  ( ph -> -. V .<_ ( Q .\/ T ) )
53 nbrne2 
 |-  ( ( C .<_ ( Q .\/ T ) /\ -. V .<_ ( Q .\/ T ) ) -> C =/= V )
54 16 52 53 syl2anc 
 |-  ( ph -> C =/= V )
55 1 4atexlemw 
 |-  ( ph -> W e. H )
56 45 55 jca 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
57 1 4atexlempw 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
58 1 4atexlems 
 |-  ( ph -> S e. A )
59 1 2 3 4 5 6 7 8 9 4atexlemc 
 |-  ( ph -> C e. A )
60 1 2 3 5 4atexlempns 
 |-  ( ph -> P =/= S )
61 13 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ S ) )
62 10 11 12 61 syl3anc 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ S ) )
63 9 62 eqbrtrid 
 |-  ( ph -> C .<_ ( P .\/ S ) )
64 2 3 4 5 6 8 lhpat3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( S e. A /\ C e. A ) /\ ( P =/= S /\ C .<_ ( P .\/ S ) ) ) -> ( -. C .<_ W <-> C =/= V ) )
65 56 57 58 59 60 63 64 syl222anc 
 |-  ( ph -> ( -. C .<_ W <-> C =/= V ) )
66 54 65 mpbird 
 |-  ( ph -> -. C .<_ W )