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Metamath Proof Explorer

Theorem prmdvdsexpb

Description: A prime divides a positive power of another iff they are equal. (Contributed by Paul Chapman, 30-Nov-2012) (Revised by Mario Carneiro, 24-Feb-2014)

Ref Expression
Assertion prmdvdsexpb 
|- ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P = Q ) )

Proof

Step Hyp Ref Expression
1 prmz 
 |-  ( Q e. Prime -> Q e. ZZ )
2 prmdvdsexp 
 |-  ( ( P e. Prime /\ Q e. ZZ /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P || Q ) )
3 1 2 syl3an2 
 |-  ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P || Q ) )
4 prmuz2 
 |-  ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
5 dvdsprm 
 |-  ( ( P e. ( ZZ>= ` 2 ) /\ Q e. Prime ) -> ( P || Q <-> P = Q ) )
6 4 5 sylan 
 |-  ( ( P e. Prime /\ Q e. Prime ) -> ( P || Q <-> P = Q ) )
7 6 3adant3 
 |-  ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || Q <-> P = Q ) )
8 3 7 bitrd 
 |-  ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P = Q ) )