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Metamath Proof Explorer

Theorem 4atexlemcnd

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 ) )
4thatlem0.d 
|- D = ( ( R .\/ T ) ./\ ( P .\/ S ) )
Assertion 4atexlemcnd 
|- ( ph -> C =/= D )

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 4thatlem0.d 
 |-  D = ( ( R .\/ T ) ./\ ( P .\/ S ) )
11 1 2 3 4 5 6 7 8 4atexlemtlw 
 |-  ( ph -> T .<_ W )
12 1 2 3 4 5 6 7 8 9 4atexlemnclw 
 |-  ( ph -> -. C .<_ W )
13 nbrne2 
 |-  ( ( T .<_ W /\ -. C .<_ W ) -> T =/= C )
14 11 12 13 syl2anc 
 |-  ( ph -> T =/= C )
15 1 4atexlemk 
 |-  ( ph -> K e. HL )
16 1 4atexlemq 
 |-  ( ph -> Q e. A )
17 1 4atexlemt 
 |-  ( ph -> T e. A )
18 3 5 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) = ( T .\/ Q ) )
19 15 16 17 18 syl3anc 
 |-  ( ph -> ( Q .\/ T ) = ( T .\/ Q ) )
20 simp221 
 |-  ( ( ( ( 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 ) ) ) -> R e. A )
21 1 20 sylbi 
 |-  ( ph -> R e. A )
22 3 5 hlatjcom 
 |-  ( ( K e. HL /\ R e. A /\ T e. A ) -> ( R .\/ T ) = ( T .\/ R ) )
23 15 21 17 22 syl3anc 
 |-  ( ph -> ( R .\/ T ) = ( T .\/ R ) )
24 19 23 oveq12d 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( R .\/ T ) ) = ( ( T .\/ Q ) ./\ ( T .\/ R ) ) )
25 1 4atexlemkc 
 |-  ( ph -> K e. CvLat )
26 1 4atexlemp 
 |-  ( ph -> P e. A )
27 1 4atexlempnq 
 |-  ( ph -> P =/= Q )
28 simp223 
 |-  ( ( ( ( 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 ) ) ) -> ( P .\/ R ) = ( Q .\/ R ) )
29 1 28 sylbi 
 |-  ( ph -> ( P .\/ R ) = ( Q .\/ R ) )
30 5 3 cvlsupr6 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R =/= Q )
31 30 necomd 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> Q =/= R )
32 25 26 16 21 27 29 31 syl132anc 
 |-  ( ph -> Q =/= R )
33 1 2 3 4 5 6 7 8 4atexlemntlpq 
 |-  ( ph -> -. T .<_ ( P .\/ Q ) )
34 5 3 cvlsupr7 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( P .\/ Q ) = ( R .\/ Q ) )
35 25 26 16 21 27 29 34 syl132anc 
 |-  ( ph -> ( P .\/ Q ) = ( R .\/ Q ) )
36 3 5 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) = ( R .\/ Q ) )
37 15 16 21 36 syl3anc 
 |-  ( ph -> ( Q .\/ R ) = ( R .\/ Q ) )
38 35 37 eqtr4d 
 |-  ( ph -> ( P .\/ Q ) = ( Q .\/ R ) )
39 38 breq2d 
 |-  ( ph -> ( T .<_ ( P .\/ Q ) <-> T .<_ ( Q .\/ R ) ) )
40 33 39 mtbid 
 |-  ( ph -> -. T .<_ ( Q .\/ R ) )
41 2 3 4 5 2llnma2 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ T e. A ) /\ ( Q =/= R /\ -. T .<_ ( Q .\/ R ) ) ) -> ( ( T .\/ Q ) ./\ ( T .\/ R ) ) = T )
42 15 16 21 17 32 40 41 syl132anc 
 |-  ( ph -> ( ( T .\/ Q ) ./\ ( T .\/ R ) ) = T )
43 24 42 eqtr2d 
 |-  ( ph -> T = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) )
44 43 adantr 
 |-  ( ( ph /\ C = D ) -> T = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) )
45 1 4atexlemkl 
 |-  ( ph -> K e. Lat )
46 1 3 5 4atexlemqtb 
 |-  ( ph -> ( Q .\/ T ) e. ( Base ` K ) )
47 1 3 5 4atexlempsb 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
48 eqid 
 |-  ( Base ` K ) = ( Base ` K )
49 48 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
50 45 46 47 49 syl3anc 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
51 9 50 eqbrtrid 
 |-  ( ph -> C .<_ ( Q .\/ T ) )
52 51 adantr 
 |-  ( ( ph /\ C = D ) -> C .<_ ( Q .\/ T ) )
53 simpr 
 |-  ( ( ph /\ C = D ) -> C = D )
54 48 3 5 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ T e. A ) -> ( R .\/ T ) e. ( Base ` K ) )
55 15 21 17 54 syl3anc 
 |-  ( ph -> ( R .\/ T ) e. ( Base ` K ) )
56 48 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( R .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( R .\/ T ) ./\ ( P .\/ S ) ) .<_ ( R .\/ T ) )
57 45 55 47 56 syl3anc 
 |-  ( ph -> ( ( R .\/ T ) ./\ ( P .\/ S ) ) .<_ ( R .\/ T ) )
58 10 57 eqbrtrid 
 |-  ( ph -> D .<_ ( R .\/ T ) )
59 58 adantr 
 |-  ( ( ph /\ C = D ) -> D .<_ ( R .\/ T ) )
60 53 59 eqbrtrd 
 |-  ( ( ph /\ C = D ) -> C .<_ ( R .\/ T ) )
61 1 2 3 4 5 6 7 8 9 4atexlemc 
 |-  ( ph -> C e. A )
62 48 5 atbase 
 |-  ( C e. A -> C e. ( Base ` K ) )
63 61 62 syl 
 |-  ( ph -> C e. ( Base ` K ) )
64 48 2 4 latlem12 
 |-  ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( R .\/ T ) e. ( Base ` K ) ) ) -> ( ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ T ) ) <-> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) )
65 45 63 46 55 64 syl13anc 
 |-  ( ph -> ( ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ T ) ) <-> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) )
66 65 adantr 
 |-  ( ( ph /\ C = D ) -> ( ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ T ) ) <-> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) )
67 52 60 66 mpbi2and 
 |-  ( ( ph /\ C = D ) -> C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) )
68 hlatl 
 |-  ( K e. HL -> K e. AtLat )
69 15 68 syl 
 |-  ( ph -> K e. AtLat )
70 43 17 eqeltrrd 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( R .\/ T ) ) e. A )
71 2 5 atcmp 
 |-  ( ( K e. AtLat /\ C e. A /\ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) e. A ) -> ( C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) <-> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) )
72 69 61 70 71 syl3anc 
 |-  ( ph -> ( C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) <-> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) )
73 72 adantr 
 |-  ( ( ph /\ C = D ) -> ( C .<_ ( ( Q .\/ T ) ./\ ( R .\/ T ) ) <-> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) ) )
74 67 73 mpbid 
 |-  ( ( ph /\ C = D ) -> C = ( ( Q .\/ T ) ./\ ( R .\/ T ) ) )
75 44 74 eqtr4d 
 |-  ( ( ph /\ C = D ) -> T = C )
76 75 ex 
 |-  ( ph -> ( C = D -> T = C ) )
77 76 necon3d 
 |-  ( ph -> ( T =/= C -> C =/= D ) )
78 14 77 mpd 
 |-  ( ph -> C =/= D )