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Metamath Proof Explorer

Theorem atcmp

Description: If two atoms are comparable, they are equal. ( atsseq analog.) (Contributed by NM, 13-Oct-2011)

Ref Expression
Hypotheses atcmp.l 
|- .<_ = ( le ` K )
atcmp.a 
|- A = ( Atoms ` K )
Assertion atcmp 
|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P .<_ Q <-> P = Q ) )

Proof

Step Hyp Ref Expression
1 atcmp.l 
 |-  .<_ = ( le ` K )
2 atcmp.a 
 |-  A = ( Atoms ` K )
3 atlpos 
 |-  ( K e. AtLat -> K e. Poset )
4 3 3ad2ant1 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> K e. Poset )
5 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 5 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
7 6 3ad2ant2 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> P e. ( Base ` K ) )
8 5 2 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
9 8 3ad2ant3 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> Q e. ( Base ` K ) )
10 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
11 5 10 atl0cl 
 |-  ( K e. AtLat -> ( 0. ` K ) e. ( Base ` K ) )
12 11 3ad2ant1 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( 0. ` K ) e. ( Base ` K ) )
13 eqid 
 |-  (
14 10 13 2 atcvr0 
 |-  ( ( K e. AtLat /\ P e. A ) -> ( 0. ` K ) (
15 14 3adant3 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( 0. ` K ) (
16 10 13 2 atcvr0 
 |-  ( ( K e. AtLat /\ Q e. A ) -> ( 0. ` K ) (
17 16 3adant2 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( 0. ` K ) (
18 5 1 13 cvrcmp 
 |-  ( ( K e. Poset /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( 0. ` K ) e. ( Base ` K ) ) /\ ( ( 0. ` K ) (  ( P .<_ Q <-> P = Q ) )
19 4 7 9 12 15 17 18 syl132anc 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P .<_ Q <-> P = Q ) )