Metamath Proof Explorer

Theorem isat3

Description: The predicate "is an atom". ( elat2 analog.) (Contributed by NM, 27-Apr-2014)

Ref Expression
Hypotheses isat3.b 
|- B = ( Base ` K )
isat3.l 
|- .<_ = ( le ` K )
isat3.z 
|- .0. = ( 0. ` K )
isat3.a 
|- A = ( Atoms ` K )
Assertion isat3 
|- ( K e. AtLat -> ( P e. A <-> ( P e. B /\ P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )

Proof

Step Hyp Ref Expression
1 isat3.b 
 |-  B = ( Base ` K )
2 isat3.l 
 |-  .<_ = ( le ` K )
3 isat3.z 
 |-  .0. = ( 0. ` K )
4 isat3.a 
 |-  A = ( Atoms ` K )
5 eqid 
 |-  (
6 1 3 5 4 isat 
 |-  ( K e. AtLat -> ( P e. A <-> ( P e. B /\ .0. (
7 simpl 
 |-  ( ( K e. AtLat /\ P e. B ) -> K e. AtLat )
8 1 3 atl0cl 
 |-  ( K e. AtLat -> .0. e. B )
9 8 adantr 
 |-  ( ( K e. AtLat /\ P e. B ) -> .0. e. B )
10 simpr 
 |-  ( ( K e. AtLat /\ P e. B ) -> P e. B )
11 eqid 
 |-  ( lt ` K ) = ( lt ` K )
12 1 2 11 5 cvrval2 
 |-  ( ( K e. AtLat /\ .0. e. B /\ P e. B ) -> ( .0. (  ( .0. ( lt ` K ) P /\ A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) ) ) )
13 7 9 10 12 syl3anc 
 |-  ( ( K e. AtLat /\ P e. B ) -> ( .0. (  ( .0. ( lt ` K ) P /\ A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) ) ) )
14 1 11 3 atlltn0 
 |-  ( ( K e. AtLat /\ P e. B ) -> ( .0. ( lt ` K ) P <-> P =/= .0. ) )
15 1 11 3 atlltn0 
 |-  ( ( K e. AtLat /\ x e. B ) -> ( .0. ( lt ` K ) x <-> x =/= .0. ) )
16 15 adantlr 
 |-  ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( .0. ( lt ` K ) x <-> x =/= .0. ) )
17 16 imbi1d 
 |-  ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( ( .0. ( lt ` K ) x -> x = P ) <-> ( x =/= .0. -> x = P ) ) )
18 17 imbi2d 
 |-  ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( ( x .<_ P -> ( .0. ( lt ` K ) x -> x = P ) ) <-> ( x .<_ P -> ( x =/= .0. -> x = P ) ) ) )
19 impexp 
 |-  ( ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> ( .0. ( lt ` K ) x -> ( x .<_ P -> x = P ) ) )
20 bi2.04 
 |-  ( ( .0. ( lt ` K ) x -> ( x .<_ P -> x = P ) ) <-> ( x .<_ P -> ( .0. ( lt ` K ) x -> x = P ) ) )
21 19 20 bitri 
 |-  ( ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> ( x .<_ P -> ( .0. ( lt ` K ) x -> x = P ) ) )
22 orcom 
 |-  ( ( x = P \/ x = .0. ) <-> ( x = .0. \/ x = P ) )
23 neor 
 |-  ( ( x = .0. \/ x = P ) <-> ( x =/= .0. -> x = P ) )
24 22 23 bitri 
 |-  ( ( x = P \/ x = .0. ) <-> ( x =/= .0. -> x = P ) )
25 24 imbi2i 
 |-  ( ( x .<_ P -> ( x = P \/ x = .0. ) ) <-> ( x .<_ P -> ( x =/= .0. -> x = P ) ) )
26 18 21 25 3bitr4g 
 |-  ( ( ( K e. AtLat /\ P e. B ) /\ x e. B ) -> ( ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> ( x .<_ P -> ( x = P \/ x = .0. ) ) ) )
27 26 ralbidva 
 |-  ( ( K e. AtLat /\ P e. B ) -> ( A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) <-> A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) )
28 14 27 anbi12d 
 |-  ( ( K e. AtLat /\ P e. B ) -> ( ( .0. ( lt ` K ) P /\ A. x e. B ( ( .0. ( lt ` K ) x /\ x .<_ P ) -> x = P ) ) <-> ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )
29 13 28 bitr2d 
 |-  ( ( K e. AtLat /\ P e. B ) -> ( ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) <-> .0. (
30 29 pm5.32da 
 |-  ( K e. AtLat -> ( ( P e. B /\ ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) <-> ( P e. B /\ .0. (
31 6 30 bitr4d 
 |-  ( K e. AtLat -> ( P e. A <-> ( P e. B /\ ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) ) )
32 3anass 
 |-  ( ( P e. B /\ P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) <-> ( P e. B /\ ( P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )
33 31 32 bitr4di 
 |-  ( K e. AtLat -> ( P e. A <-> ( P e. B /\ P =/= .0. /\ A. x e. B ( x .<_ P -> ( x = P \/ x = .0. ) ) ) ) )