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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 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) ↔ 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) )

Proof

Step Hyp Ref Expression
1 prmz ⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℤ )
2 1 adantr ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → 𝑃 ∈ ℤ )
3 zexpcl ⊢ ( ( 𝑃 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑃 ↑ 𝑛 ) ∈ ℤ )
4 2 3 sylan ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑃 ↑ 𝑛 ) ∈ ℤ )
5 iddvds ⊢ ( ( 𝑃 ↑ 𝑛 ) ∈ ℤ → ( 𝑃 ↑ 𝑛 ) ∥ ( 𝑃 ↑ 𝑛 ) )
6 4 5 syl ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑃 ↑ 𝑛 ) ∥ ( 𝑃 ↑ 𝑛 ) )
7 breq1 ⊢ ( 𝐴 = ( 𝑃 ↑ 𝑛 ) → ( 𝐴 ∥ ( 𝑃 ↑ 𝑛 ) ↔ ( 𝑃 ↑ 𝑛 ) ∥ ( 𝑃 ↑ 𝑛 ) ) )
8 6 7 syl5ibrcom ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐴 = ( 𝑃 ↑ 𝑛 ) → 𝐴 ∥ ( 𝑃 ↑ 𝑛 ) ) )
9 8 reximdva ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) → ∃ 𝑛 ∈ ℕ0 𝐴 ∥ ( 𝑃 ↑ 𝑛 ) ) )
10 pcprmpw2 ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑛 ∈ ℕ0 𝐴 ∥ ( 𝑃 ↑ 𝑛 ) ↔ 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) )
11 9 10 sylibd ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) → 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) )
12 pccl ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( 𝑃 pCnt 𝐴 ) ∈ ℕ0 )
13 oveq2 ⊢ ( 𝑛 = ( 𝑃 pCnt 𝐴 ) → ( 𝑃 ↑ 𝑛 ) = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) )
14 13 rspceeqv ⊢ ( ( ( 𝑃 pCnt 𝐴 ) ∈ ℕ0 ∧ 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) → ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) )
15 14 ex ⊢ ( ( 𝑃 pCnt 𝐴 ) ∈ ℕ0 → ( 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) → ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) ) )
16 12 15 syl ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) → ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) ) )
17 11 16 impbid ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ ) → ( ∃ 𝑛 ∈ ℕ0 𝐴 = ( 𝑃 ↑ 𝑛 ) ↔ 𝐴 = ( 𝑃 ↑ ( 𝑃 pCnt 𝐴 ) ) ) )