Metamath Proof Explorer


Theorem expne0d

Description: Nonnegative integer exponentiation is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1
|- ( ph -> A e. CC )
sqrecd.1
|- ( ph -> A =/= 0 )
expclzd.3
|- ( ph -> N e. ZZ )
Assertion expne0d
|- ( ph -> ( A ^ N ) =/= 0 )

Proof

Step Hyp Ref Expression
1 expcld.1
 |-  ( ph -> A e. CC )
2 sqrecd.1
 |-  ( ph -> A =/= 0 )
3 expclzd.3
 |-  ( ph -> N e. ZZ )
4 expne0i
 |-  ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) =/= 0 )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( A ^ N ) =/= 0 )