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Metamath Proof Explorer

Theorem leatb

Description: A poset element less than or equal to an atom equals either zero or the atom. ( atss analog.) (Contributed by NM, 17-Nov-2011)

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

Proof

Step Hyp Ref Expression
1 leatom.b 
 |-  B = ( Base ` K )
2 leatom.l 
 |-  .<_ = ( le ` K )
3 leatom.z 
 |-  .0. = ( 0. ` K )
4 leatom.a 
 |-  A = ( Atoms ` K )
5 1 2 3 op0le 
 |-  ( ( K e. OP /\ X e. B ) -> .0. .<_ X )
6 5 3adant3 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> .0. .<_ X )
7 6 biantrurd 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( .0. .<_ X /\ X .<_ P ) ) )
8 opposet 
 |-  ( K e. OP -> K e. Poset )
9 8 3ad2ant1 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> K e. Poset )
10 1 3 op0cl 
 |-  ( K e. OP -> .0. e. B )
11 1 4 atbase 
 |-  ( P e. A -> P e. B )
12 id 
 |-  ( X e. B -> X e. B )
13 10 11 12 3anim123i 
 |-  ( ( K e. OP /\ P e. A /\ X e. B ) -> ( .0. e. B /\ P e. B /\ X e. B ) )
14 13 3com23 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( .0. e. B /\ P e. B /\ X e. B ) )
15 eqid 
 |-  (
16 3 15 4 atcvr0 
 |-  ( ( K e. OP /\ P e. A ) -> .0. (
17 16 3adant2 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> .0. (
18 1 2 15 cvrnbtwn4 
 |-  ( ( K e. Poset /\ ( .0. e. B /\ P e. B /\ X e. B ) /\ .0. (  ( ( .0. .<_ X /\ X .<_ P ) <-> ( .0. = X \/ X = P ) ) )
19 9 14 17 18 syl3anc 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( ( .0. .<_ X /\ X .<_ P ) <-> ( .0. = X \/ X = P ) ) )
20 eqcom 
 |-  ( .0. = X <-> X = .0. )
21 20 orbi1i 
 |-  ( ( .0. = X \/ X = P ) <-> ( X = .0. \/ X = P ) )
22 19 21 bitrdi 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( ( .0. .<_ X /\ X .<_ P ) <-> ( X = .0. \/ X = P ) ) )
23 7 22 bitrd 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X = .0. \/ X = P ) ) )
24 orcom 
 |-  ( ( X = .0. \/ X = P ) <-> ( X = P \/ X = .0. ) )
25 23 24 bitrdi 
 |-  ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X = P \/ X = .0. ) ) )