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Metamath Proof Explorer

Theorem atnle

Description: Two ways of expressing "an atom is not less than or equal to a lattice element." ( atnssm0 analog.) (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses atnle.b 
|- B = ( Base ` K )
atnle.l 
|- .<_ = ( le ` K )
atnle.m 
|- ./\ = ( meet ` K )
atnle.z 
|- .0. = ( 0. ` K )
atnle.a 
|- A = ( Atoms ` K )
Assertion atnle 
|- ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( -. P .<_ X <-> ( P ./\ X ) = .0. ) )

Proof

Step Hyp Ref Expression
1 atnle.b 
 |-  B = ( Base ` K )
2 atnle.l 
 |-  .<_ = ( le ` K )
3 atnle.m 
 |-  ./\ = ( meet ` K )
4 atnle.z 
 |-  .0. = ( 0. ` K )
5 atnle.a 
 |-  A = ( Atoms ` K )
6 simpl1 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) -> K e. AtLat )
7 atllat 
 |-  ( K e. AtLat -> K e. Lat )
8 7 3ad2ant1 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> K e. Lat )
9 1 5 atbase 
 |-  ( P e. A -> P e. B )
10 9 3ad2ant2 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> P e. B )
11 simp3 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> X e. B )
12 1 3 latmcl 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> ( P ./\ X ) e. B )
13 8 10 11 12 syl3anc 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( P ./\ X ) e. B )
14 13 adantr 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) -> ( P ./\ X ) e. B )
15 simpr 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) -> ( P ./\ X ) =/= .0. )
16 1 2 4 5 atlex 
 |-  ( ( K e. AtLat /\ ( P ./\ X ) e. B /\ ( P ./\ X ) =/= .0. ) -> E. y e. A y .<_ ( P ./\ X ) )
17 6 14 15 16 syl3anc 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) -> E. y e. A y .<_ ( P ./\ X ) )
18 simpl1 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> K e. AtLat )
19 18 7 syl 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> K e. Lat )
20 1 5 atbase 
 |-  ( y e. A -> y e. B )
21 20 adantl 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> y e. B )
22 simpl2 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> P e. A )
23 22 9 syl 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> P e. B )
24 simpl3 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> X e. B )
25 1 2 3 latlem12 
 |-  ( ( K e. Lat /\ ( y e. B /\ P e. B /\ X e. B ) ) -> ( ( y .<_ P /\ y .<_ X ) <-> y .<_ ( P ./\ X ) ) )
26 19 21 23 24 25 syl13anc 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> ( ( y .<_ P /\ y .<_ X ) <-> y .<_ ( P ./\ X ) ) )
27 simpr 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> y e. A )
28 2 5 atcmp 
 |-  ( ( K e. AtLat /\ y e. A /\ P e. A ) -> ( y .<_ P <-> y = P ) )
29 18 27 22 28 syl3anc 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> ( y .<_ P <-> y = P ) )
30 breq1 
 |-  ( y = P -> ( y .<_ X <-> P .<_ X ) )
31 30 biimpd 
 |-  ( y = P -> ( y .<_ X -> P .<_ X ) )
32 29 31 biimtrdi 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> ( y .<_ P -> ( y .<_ X -> P .<_ X ) ) )
33 32 impd 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> ( ( y .<_ P /\ y .<_ X ) -> P .<_ X ) )
34 26 33 sylbird 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ y e. A ) -> ( y .<_ ( P ./\ X ) -> P .<_ X ) )
35 34 adantlr 
 |-  ( ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) /\ y e. A ) -> ( y .<_ ( P ./\ X ) -> P .<_ X ) )
36 35 rexlimdva 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) -> ( E. y e. A y .<_ ( P ./\ X ) -> P .<_ X ) )
37 17 36 mpd 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) =/= .0. ) -> P .<_ X )
38 37 ex 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( ( P ./\ X ) =/= .0. -> P .<_ X ) )
39 38 necon1bd 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( -. P .<_ X -> ( P ./\ X ) = .0. ) )
40 4 5 atn0 
 |-  ( ( K e. AtLat /\ P e. A ) -> P =/= .0. )
41 40 3adant3 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> P =/= .0. )
42 1 2 3 latleeqm1 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> ( P .<_ X <-> ( P ./\ X ) = P ) )
43 8 10 11 42 syl3anc 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( P .<_ X <-> ( P ./\ X ) = P ) )
44 43 adantr 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ X <-> ( P ./\ X ) = P ) )
45 eqeq1 
 |-  ( ( P ./\ X ) = P -> ( ( P ./\ X ) = .0. <-> P = .0. ) )
46 45 biimpcd 
 |-  ( ( P ./\ X ) = .0. -> ( ( P ./\ X ) = P -> P = .0. ) )
47 46 adantl 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( ( P ./\ X ) = P -> P = .0. ) )
48 44 47 sylbid 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P .<_ X -> P = .0. ) )
49 48 necon3ad 
 |-  ( ( ( K e. AtLat /\ P e. A /\ X e. B ) /\ ( P ./\ X ) = .0. ) -> ( P =/= .0. -> -. P .<_ X ) )
50 49 ex 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( ( P ./\ X ) = .0. -> ( P =/= .0. -> -. P .<_ X ) ) )
51 41 50 mpid 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( ( P ./\ X ) = .0. -> -. P .<_ X ) )
52 39 51 impbid 
 |-  ( ( K e. AtLat /\ P e. A /\ X e. B ) -> ( -. P .<_ X <-> ( P ./\ X ) = .0. ) )