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Metamath Proof Explorer

Theorem 4atexlemc

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 4atexlemc 
|- ( ph -> C e. A )

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 4 latmcom 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
15 10 11 12 14 syl3anc 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
16 9 15 eqtrid 
 |-  ( ph -> C = ( ( P .\/ S ) ./\ ( Q .\/ T ) ) )
17 1 4atexlemk 
 |-  ( ph -> K e. HL )
18 1 4atexlemp 
 |-  ( ph -> P e. A )
19 1 4atexlems 
 |-  ( ph -> S e. A )
20 1 4atexlemq 
 |-  ( ph -> Q e. A )
21 1 4atexlemt 
 |-  ( ph -> T e. A )
22 1 2 3 5 4atexlempns 
 |-  ( ph -> P =/= S )
23 1 2 3 4 5 6 7 8 4atexlemntlpq 
 |-  ( ph -> -. T .<_ ( P .\/ Q ) )
24 2 3 5 atnlej2 
 |-  ( ( K e. HL /\ ( T e. A /\ P e. A /\ Q e. A ) /\ -. T .<_ ( P .\/ Q ) ) -> T =/= Q )
25 24 necomd 
 |-  ( ( K e. HL /\ ( T e. A /\ P e. A /\ Q e. A ) /\ -. T .<_ ( P .\/ Q ) ) -> Q =/= T )
26 17 21 18 20 23 25 syl131anc 
 |-  ( ph -> Q =/= T )
27 1 4atexlempnq 
 |-  ( ph -> P =/= Q )
28 1 4atexlemnslpq 
 |-  ( ph -> -. S .<_ ( P .\/ Q ) )
29 2 3 5 4atlem0ae 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> -. Q .<_ ( P .\/ S ) )
30 17 18 20 19 27 28 29 syl132anc 
 |-  ( ph -> -. Q .<_ ( P .\/ S ) )
31 13 5 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
32 21 31 syl 
 |-  ( ph -> T e. ( Base ` K ) )
33 1 2 3 4 5 6 7 4atexlemu 
 |-  ( ph -> U e. A )
34 1 2 3 4 5 6 7 8 4atexlemv 
 |-  ( ph -> V e. A )
35 13 3 5 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ V e. A ) -> ( U .\/ V ) e. ( Base ` K ) )
36 17 33 34 35 syl3anc 
 |-  ( ph -> ( U .\/ V ) e. ( Base ` K ) )
37 13 5 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
38 20 37 syl 
 |-  ( ph -> Q e. ( Base ` K ) )
39 13 3 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) )
40 10 12 38 39 syl3anc 
 |-  ( ph -> ( ( P .\/ S ) .\/ Q ) e. ( Base ` K ) )
41 1 4atexlemkc 
 |-  ( ph -> K e. CvLat )
42 1 2 3 4 5 6 7 8 4atexlemunv 
 |-  ( ph -> U =/= V )
43 1 4atexlemutvt 
 |-  ( ph -> ( U .\/ T ) = ( V .\/ T ) )
44 5 2 3 cvlsupr4 
 |-  ( ( K e. CvLat /\ ( U e. A /\ V e. A /\ T e. A ) /\ ( U =/= V /\ ( U .\/ T ) = ( V .\/ T ) ) ) -> T .<_ ( U .\/ V ) )
45 41 33 34 21 42 43 44 syl132anc 
 |-  ( ph -> T .<_ ( U .\/ V ) )
46 13 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
47 17 18 20 46 syl3anc 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
48 1 6 4atexlemwb 
 |-  ( ph -> W e. ( Base ` K ) )
49 13 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
50 10 47 48 49 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
51 7 50 eqbrtrid 
 |-  ( ph -> U .<_ ( P .\/ Q ) )
52 13 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
53 10 12 48 52 syl3anc 
 |-  ( ph -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
54 8 53 eqbrtrid 
 |-  ( ph -> V .<_ ( P .\/ S ) )
55 13 5 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
56 33 55 syl 
 |-  ( ph -> U e. ( Base ` K ) )
57 13 5 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
58 34 57 syl 
 |-  ( ph -> V e. ( Base ` K ) )
59 13 2 3 latjlej12 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ ( V e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) ) -> ( ( U .<_ ( P .\/ Q ) /\ V .<_ ( P .\/ S ) ) -> ( U .\/ V ) .<_ ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) )
60 10 56 47 58 12 59 syl122anc 
 |-  ( ph -> ( ( U .<_ ( P .\/ Q ) /\ V .<_ ( P .\/ S ) ) -> ( U .\/ V ) .<_ ( ( P .\/ Q ) .\/ ( P .\/ S ) ) ) )
61 51 54 60 mp2and 
 |-  ( ph -> ( U .\/ V ) .<_ ( ( P .\/ Q ) .\/ ( P .\/ S ) ) )
62 3 5 hlatjass 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .\/ S ) = ( P .\/ ( Q .\/ S ) ) )
63 17 18 20 19 62 syl13anc 
 |-  ( ph -> ( ( P .\/ Q ) .\/ S ) = ( P .\/ ( Q .\/ S ) ) )
64 13 5 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
65 18 64 syl 
 |-  ( ph -> P e. ( Base ` K ) )
66 13 5 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
67 19 66 syl 
 |-  ( ph -> S e. ( Base ` K ) )
68 13 3 latj32 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ S ) = ( ( P .\/ S ) .\/ Q ) )
69 10 65 38 67 68 syl13anc 
 |-  ( ph -> ( ( P .\/ Q ) .\/ S ) = ( ( P .\/ S ) .\/ Q ) )
70 13 3 latjjdi 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( P .\/ ( Q .\/ S ) ) = ( ( P .\/ Q ) .\/ ( P .\/ S ) ) )
71 10 65 38 67 70 syl13anc 
 |-  ( ph -> ( P .\/ ( Q .\/ S ) ) = ( ( P .\/ Q ) .\/ ( P .\/ S ) ) )
72 63 69 71 3eqtr3rd 
 |-  ( ph -> ( ( P .\/ Q ) .\/ ( P .\/ S ) ) = ( ( P .\/ S ) .\/ Q ) )
73 61 72 breqtrd 
 |-  ( ph -> ( U .\/ V ) .<_ ( ( P .\/ S ) .\/ Q ) )
74 13 2 10 32 36 40 45 73 lattrd 
 |-  ( ph -> T .<_ ( ( P .\/ S ) .\/ Q ) )
75 2 3 4 5 2atmat 
 |-  ( ( ( K e. HL /\ P e. A /\ S e. A ) /\ ( Q e. A /\ T e. A /\ P =/= S ) /\ ( Q =/= T /\ -. Q .<_ ( P .\/ S ) /\ T .<_ ( ( P .\/ S ) .\/ Q ) ) ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A )
76 17 18 19 20 21 22 26 30 74 75 syl333anc 
 |-  ( ph -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A )
77 16 76 eqeltrd 
 |-  ( ph -> C e. A )