Metamath Proof Explorer

Theorem 4atexlemtlw

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 )
Assertion 4atexlemtlw 
|- ( ph -> T .<_ 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 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 1 4atexlemkl 
 |-  ( ph -> K e. Lat )
11 1 4atexlemt 
 |-  ( ph -> T e. A )
12 9 5 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
13 11 12 syl 
 |-  ( ph -> T e. ( Base ` K ) )
14 1 4atexlemk 
 |-  ( ph -> K e. HL )
15 1 2 3 4 5 6 7 4atexlemu 
 |-  ( ph -> U e. A )
16 1 2 3 4 5 6 7 8 4atexlemv 
 |-  ( ph -> V e. A )
17 9 3 5 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ V e. A ) -> ( U .\/ V ) e. ( Base ` K ) )
18 14 15 16 17 syl3anc 
 |-  ( ph -> ( U .\/ V ) e. ( Base ` K ) )
19 1 6 4atexlemwb 
 |-  ( ph -> W e. ( Base ` K ) )
20 1 4atexlemkc 
 |-  ( ph -> K e. CvLat )
21 1 2 3 4 5 6 7 8 4atexlemunv 
 |-  ( ph -> U =/= V )
22 1 4atexlemutvt 
 |-  ( ph -> ( U .\/ T ) = ( V .\/ T ) )
23 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 ) )
24 20 15 16 11 21 22 23 syl132anc 
 |-  ( ph -> T .<_ ( U .\/ V ) )
25 1 4atexlemp 
 |-  ( ph -> P e. A )
26 1 4atexlemq 
 |-  ( ph -> Q e. A )
27 9 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
28 14 25 26 27 syl3anc 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
29 9 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
30 10 28 19 29 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ W )
31 7 30 eqbrtrid 
 |-  ( ph -> U .<_ W )
32 1 3 5 4atexlempsb 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
33 9 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
34 10 32 19 33 syl3anc 
 |-  ( ph -> ( ( P .\/ S ) ./\ W ) .<_ W )
35 8 34 eqbrtrid 
 |-  ( ph -> V .<_ W )
36 9 5 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
37 15 36 syl 
 |-  ( ph -> U e. ( Base ` K ) )
38 9 5 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
39 16 38 syl 
 |-  ( ph -> V e. ( Base ` K ) )
40 9 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( U e. ( Base ` K ) /\ V e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( U .<_ W /\ V .<_ W ) <-> ( U .\/ V ) .<_ W ) )
41 10 37 39 19 40 syl13anc 
 |-  ( ph -> ( ( U .<_ W /\ V .<_ W ) <-> ( U .\/ V ) .<_ W ) )
42 31 35 41 mpbi2and 
 |-  ( ph -> ( U .\/ V ) .<_ W )
43 9 2 10 13 18 19 24 42 lattrd 
 |-  ( ph -> T .<_ W )