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Metamath Proof Explorer

Theorem pcprmpw

Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015)

Ref Expression
Assertion pcprmpw 
|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A = ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )

Proof

Step Hyp Ref Expression
1 prmz 
 |-  ( P e. Prime -> P e. ZZ )
2 1 adantr 
 |-  ( ( P e. Prime /\ A e. NN ) -> P e. ZZ )
3 zexpcl 
 |-  ( ( P e. ZZ /\ n e. NN0 ) -> ( P ^ n ) e. ZZ )
4 2 3 sylan 
 |-  ( ( ( P e. Prime /\ A e. NN ) /\ n e. NN0 ) -> ( P ^ n ) e. ZZ )
5 iddvds 
 |-  ( ( P ^ n ) e. ZZ -> ( P ^ n ) || ( P ^ n ) )
6 4 5 syl 
 |-  ( ( ( P e. Prime /\ A e. NN ) /\ n e. NN0 ) -> ( P ^ n ) || ( P ^ n ) )
7 breq1 
 |-  ( A = ( P ^ n ) -> ( A || ( P ^ n ) <-> ( P ^ n ) || ( P ^ n ) ) )
8 6 7 syl5ibrcom 
 |-  ( ( ( P e. Prime /\ A e. NN ) /\ n e. NN0 ) -> ( A = ( P ^ n ) -> A || ( P ^ n ) ) )
9 8 reximdva 
 |-  ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A = ( P ^ n ) -> E. n e. NN0 A || ( P ^ n ) ) )
10 pcprmpw2 
 |-  ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )
11 9 10 sylibd 
 |-  ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A = ( P ^ n ) -> A = ( P ^ ( P pCnt A ) ) ) )
12 pccl 
 |-  ( ( P e. Prime /\ A e. NN ) -> ( P pCnt A ) e. NN0 )
13 oveq2 
 |-  ( n = ( P pCnt A ) -> ( P ^ n ) = ( P ^ ( P pCnt A ) ) )
14 13 rspceeqv 
 |-  ( ( ( P pCnt A ) e. NN0 /\ A = ( P ^ ( P pCnt A ) ) ) -> E. n e. NN0 A = ( P ^ n ) )
15 14 ex 
 |-  ( ( P pCnt A ) e. NN0 -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A = ( P ^ n ) ) )
16 12 15 syl 
 |-  ( ( P e. Prime /\ A e. NN ) -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A = ( P ^ n ) ) )
17 11 16 impbid 
 |-  ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A = ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )