Metamath Proof Explorer

Theorem 4atexlemunv

Description: Lemma for 4atexlem7 . (Contributed by NM, 21-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 4atexlemunv 
|- ( ph -> U =/= V )

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 1 4atexlemnslpq 
 |-  ( ph -> -. S .<_ ( P .\/ Q ) )
10 1 4atexlemk 
 |-  ( ph -> K e. HL )
11 1 4atexlemp 
 |-  ( ph -> P e. A )
12 1 4atexlems 
 |-  ( ph -> S e. A )
13 2 3 5 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> S .<_ ( P .\/ S ) )
14 10 11 12 13 syl3anc 
 |-  ( ph -> S .<_ ( P .\/ S ) )
15 14 adantr 
 |-  ( ( ph /\ U = V ) -> S .<_ ( P .\/ S ) )
16 1 4atexlemkl 
 |-  ( ph -> K e. Lat )
17 1 3 5 4atexlempsb 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
18 1 6 4atexlemwb 
 |-  ( ph -> W e. ( Base ` K ) )
19 eqid 
 |-  ( Base ` K ) = ( Base ` K )
20 19 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
21 16 17 18 20 syl3anc 
 |-  ( ph -> ( ( P .\/ S ) ./\ W ) .<_ ( P .\/ S ) )
22 8 21 eqbrtrid 
 |-  ( ph -> V .<_ ( P .\/ S ) )
23 1 4atexlemkc 
 |-  ( ph -> K e. CvLat )
24 1 2 3 4 5 6 7 8 4atexlemv 
 |-  ( ph -> V e. A )
25 19 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ S ) ./\ W ) .<_ W )
26 16 17 18 25 syl3anc 
 |-  ( ph -> ( ( P .\/ S ) ./\ W ) .<_ W )
27 8 26 eqbrtrid 
 |-  ( ph -> V .<_ W )
28 1 4atexlempw 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
29 28 simprd 
 |-  ( ph -> -. P .<_ W )
30 nbrne2 
 |-  ( ( V .<_ W /\ -. P .<_ W ) -> V =/= P )
31 27 29 30 syl2anc 
 |-  ( ph -> V =/= P )
32 2 3 5 cvlatexchb1 
 |-  ( ( K e. CvLat /\ ( V e. A /\ S e. A /\ P e. A ) /\ V =/= P ) -> ( V .<_ ( P .\/ S ) <-> ( P .\/ V ) = ( P .\/ S ) ) )
33 23 24 12 11 31 32 syl131anc 
 |-  ( ph -> ( V .<_ ( P .\/ S ) <-> ( P .\/ V ) = ( P .\/ S ) ) )
34 22 33 mpbid 
 |-  ( ph -> ( P .\/ V ) = ( P .\/ S ) )
35 34 adantr 
 |-  ( ( ph /\ U = V ) -> ( P .\/ V ) = ( P .\/ S ) )
36 oveq2 
 |-  ( U = V -> ( P .\/ U ) = ( P .\/ V ) )
37 36 eqcomd 
 |-  ( U = V -> ( P .\/ V ) = ( P .\/ U ) )
38 1 4atexlemq 
 |-  ( ph -> Q e. A )
39 19 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
40 10 11 38 39 syl3anc 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
41 19 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
42 16 40 18 41 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
43 7 42 eqbrtrid 
 |-  ( ph -> U .<_ ( P .\/ Q ) )
44 1 2 3 4 5 6 7 4atexlemu 
 |-  ( ph -> U e. A )
45 19 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
46 16 40 18 45 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ W )
47 7 46 eqbrtrid 
 |-  ( ph -> U .<_ W )
48 nbrne2 
 |-  ( ( U .<_ W /\ -. P .<_ W ) -> U =/= P )
49 47 29 48 syl2anc 
 |-  ( ph -> U =/= P )
50 2 3 5 cvlatexchb1 
 |-  ( ( K e. CvLat /\ ( U e. A /\ Q e. A /\ P e. A ) /\ U =/= P ) -> ( U .<_ ( P .\/ Q ) <-> ( P .\/ U ) = ( P .\/ Q ) ) )
51 23 44 38 11 49 50 syl131anc 
 |-  ( ph -> ( U .<_ ( P .\/ Q ) <-> ( P .\/ U ) = ( P .\/ Q ) ) )
52 43 51 mpbid 
 |-  ( ph -> ( P .\/ U ) = ( P .\/ Q ) )
53 37 52 sylan9eqr 
 |-  ( ( ph /\ U = V ) -> ( P .\/ V ) = ( P .\/ Q ) )
54 35 53 eqtr3d 
 |-  ( ( ph /\ U = V ) -> ( P .\/ S ) = ( P .\/ Q ) )
55 15 54 breqtrd 
 |-  ( ( ph /\ U = V ) -> S .<_ ( P .\/ Q ) )
56 55 ex 
 |-  ( ph -> ( U = V -> S .<_ ( P .\/ Q ) ) )
57 56 necon3bd 
 |-  ( ph -> ( -. S .<_ ( P .\/ Q ) -> U =/= V ) )
58 9 57 mpd 
 |-  ( ph -> U =/= V )