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Metamath Proof Explorer

Theorem 0ltat

Description: An atom is greater than zero. (Contributed by NM, 4-Jul-2012)

Ref Expression
Hypotheses 0ltat.z 
|- .0. = ( 0. ` K )
0ltat.s 
|- .< = ( lt ` K )
0ltat.a 
|- A = ( Atoms ` K )
Assertion 0ltat 
|- ( ( K e. OP /\ P e. A ) -> .0. .< P )

Proof

Step Hyp Ref Expression
1 0ltat.z 
 |-  .0. = ( 0. ` K )
2 0ltat.s 
 |-  .< = ( lt ` K )
3 0ltat.a 
 |-  A = ( Atoms ` K )
4 simpl 
 |-  ( ( K e. OP /\ P e. A ) -> K e. OP )
5 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 5 1 op0cl 
 |-  ( K e. OP -> .0. e. ( Base ` K ) )
7 6 adantr 
 |-  ( ( K e. OP /\ P e. A ) -> .0. e. ( Base ` K ) )
8 5 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
9 8 adantl 
 |-  ( ( K e. OP /\ P e. A ) -> P e. ( Base ` K ) )
10 eqid 
 |-  (
11 1 10 3 atcvr0 
 |-  ( ( K e. OP /\ P e. A ) -> .0. (
12 5 2 10 cvrlt 
 |-  ( ( ( K e. OP /\ .0. e. ( Base ` K ) /\ P e. ( Base ` K ) ) /\ .0. (  .0. .< P )
13 4 7 9 11 12 syl31anc 
 |-  ( ( K e. OP /\ P e. A ) -> .0. .< P )