Metamath Proof Explorer

Theorem 4atexlemex2

Description: Lemma for 4atexlem7 . Show that when C =/= S , C satisfies the existence condition of the consequent. (Contributed by NM, 25-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 4atexlemex2 
|- ( ( ph /\ C =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )

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 2 3 4 5 6 7 8 9 4atexlemc 
 |-  ( ph -> C e. A )
11 10 adantr 
 |-  ( ( ph /\ C =/= S ) -> C e. A )
12 1 2 3 4 5 6 7 8 9 4atexlemnclw 
 |-  ( ph -> -. C .<_ W )
13 12 adantr 
 |-  ( ( ph /\ C =/= S ) -> -. C .<_ W )
14 1 2 3 4 5 6 7 8 4atexlemntlpq 
 |-  ( ph -> -. T .<_ ( P .\/ Q ) )
15 id 
 |-  ( C = P -> C = P )
16 9 15 eqtr3id 
 |-  ( C = P -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = P )
17 16 adantl 
 |-  ( ( ph /\ C = P ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) = P )
18 1 4atexlemkl 
 |-  ( ph -> K e. Lat )
19 1 3 5 4atexlemqtb 
 |-  ( ph -> ( Q .\/ T ) e. ( Base ` K ) )
20 1 3 5 4atexlempsb 
 |-  ( ph -> ( P .\/ S ) e. ( Base ` K ) )
21 eqid 
 |-  ( Base ` K ) = ( Base ` K )
22 21 2 4 latmle1 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
23 18 19 20 22 syl3anc 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( Q .\/ T ) )
24 1 4atexlemk 
 |-  ( ph -> K e. HL )
25 1 4atexlemq 
 |-  ( ph -> Q e. A )
26 1 4atexlemt 
 |-  ( ph -> T e. A )
27 3 5 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ T e. A ) -> ( Q .\/ T ) = ( T .\/ Q ) )
28 24 25 26 27 syl3anc 
 |-  ( ph -> ( Q .\/ T ) = ( T .\/ Q ) )
29 23 28 breqtrd 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( T .\/ Q ) )
30 29 adantr 
 |-  ( ( ph /\ C = P ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( T .\/ Q ) )
31 17 30 eqbrtrrd 
 |-  ( ( ph /\ C = P ) -> P .<_ ( T .\/ Q ) )
32 1 4atexlemkc 
 |-  ( ph -> K e. CvLat )
33 1 4atexlemp 
 |-  ( ph -> P e. A )
34 1 4atexlempnq 
 |-  ( ph -> P =/= Q )
35 2 3 5 cvlatexch2 
 |-  ( ( K e. CvLat /\ ( P e. A /\ T e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( T .\/ Q ) -> T .<_ ( P .\/ Q ) ) )
36 32 33 26 25 34 35 syl131anc 
 |-  ( ph -> ( P .<_ ( T .\/ Q ) -> T .<_ ( P .\/ Q ) ) )
37 36 adantr 
 |-  ( ( ph /\ C = P ) -> ( P .<_ ( T .\/ Q ) -> T .<_ ( P .\/ Q ) ) )
38 31 37 mpd 
 |-  ( ( ph /\ C = P ) -> T .<_ ( P .\/ Q ) )
39 38 ex 
 |-  ( ph -> ( C = P -> T .<_ ( P .\/ Q ) ) )
40 39 necon3bd 
 |-  ( ph -> ( -. T .<_ ( P .\/ Q ) -> C =/= P ) )
41 14 40 mpd 
 |-  ( ph -> C =/= P )
42 41 adantr 
 |-  ( ( ph /\ C =/= S ) -> C =/= P )
43 simpr 
 |-  ( ( ph /\ C =/= S ) -> C =/= S )
44 21 2 4 latmle2 
 |-  ( ( K e. Lat /\ ( Q .\/ T ) e. ( Base ` K ) /\ ( P .\/ S ) e. ( Base ` K ) ) -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ S ) )
45 18 19 20 44 syl3anc 
 |-  ( ph -> ( ( Q .\/ T ) ./\ ( P .\/ S ) ) .<_ ( P .\/ S ) )
46 9 45 eqbrtrid 
 |-  ( ph -> C .<_ ( P .\/ S ) )
47 46 adantr 
 |-  ( ( ph /\ C =/= S ) -> C .<_ ( P .\/ S ) )
48 1 4atexlems 
 |-  ( ph -> S e. A )
49 1 2 3 5 4atexlempns 
 |-  ( ph -> P =/= S )
50 5 2 3 cvlsupr2 
 |-  ( ( K e. CvLat /\ ( P e. A /\ S e. A /\ C e. A ) /\ P =/= S ) -> ( ( P .\/ C ) = ( S .\/ C ) <-> ( C =/= P /\ C =/= S /\ C .<_ ( P .\/ S ) ) ) )
51 32 33 48 10 49 50 syl131anc 
 |-  ( ph -> ( ( P .\/ C ) = ( S .\/ C ) <-> ( C =/= P /\ C =/= S /\ C .<_ ( P .\/ S ) ) ) )
52 51 adantr 
 |-  ( ( ph /\ C =/= S ) -> ( ( P .\/ C ) = ( S .\/ C ) <-> ( C =/= P /\ C =/= S /\ C .<_ ( P .\/ S ) ) ) )
53 42 43 47 52 mpbir3and 
 |-  ( ( ph /\ C =/= S ) -> ( P .\/ C ) = ( S .\/ C ) )
54 breq1 
 |-  ( z = C -> ( z .<_ W <-> C .<_ W ) )
55 54 notbid 
 |-  ( z = C -> ( -. z .<_ W <-> -. C .<_ W ) )
56 oveq2 
 |-  ( z = C -> ( P .\/ z ) = ( P .\/ C ) )
57 oveq2 
 |-  ( z = C -> ( S .\/ z ) = ( S .\/ C ) )
58 56 57 eqeq12d 
 |-  ( z = C -> ( ( P .\/ z ) = ( S .\/ z ) <-> ( P .\/ C ) = ( S .\/ C ) ) )
59 55 58 anbi12d 
 |-  ( z = C -> ( ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) <-> ( -. C .<_ W /\ ( P .\/ C ) = ( S .\/ C ) ) ) )
60 59 rspcev 
 |-  ( ( C e. A /\ ( -. C .<_ W /\ ( P .\/ C ) = ( S .\/ C ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
61 11 13 53 60 syl12anc 
 |-  ( ( ph /\ C =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )