Metamath Proof Explorer

Theorem prmexpb

Description: Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012)

Ref Expression
Assertion prmexpb ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ) → ( ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ↔ ( 𝑃 = 𝑄 ∧ 𝑀 = 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 prmz ⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℤ )
2 1 adantr ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → 𝑃 ∈ ℤ )
3 2 3ad2ant1 ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑃 ∈ ℤ )
4 simp2l ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑀 ∈ ℕ )
5 iddvdsexp ⊢ ( ( 𝑃 ∈ ℤ ∧ 𝑀 ∈ ℕ ) → 𝑃 ∥ ( 𝑃 ↑ 𝑀 ) )
6 3 4 5 syl2anc ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑃 ∥ ( 𝑃 ↑ 𝑀 ) )
7 breq2 ⊢ ( ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) → ( 𝑃 ∥ ( 𝑃 ↑ 𝑀 ) ↔ 𝑃 ∥ ( 𝑄 ↑ 𝑁 ) ) )
8 7 3ad2ant3 ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 ∥ ( 𝑃 ↑ 𝑀 ) ↔ 𝑃 ∥ ( 𝑄 ↑ 𝑁 ) ) )
9 simp1l ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑃 ∈ ℙ )
10 simp1r ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑄 ∈ ℙ )
11 simp2r ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑁 ∈ ℕ )
12 prmdvdsexpb ⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑁 ∈ ℕ ) → ( 𝑃 ∥ ( 𝑄 ↑ 𝑁 ) ↔ 𝑃 = 𝑄 ) )
13 9 10 11 12 syl3anc ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 ∥ ( 𝑄 ↑ 𝑁 ) ↔ 𝑃 = 𝑄 ) )
14 8 13 bitrd ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 ∥ ( 𝑃 ↑ 𝑀 ) ↔ 𝑃 = 𝑄 ) )
15 6 14 mpbid ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑃 = 𝑄 )
16 3 zred ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑃 ∈ ℝ )
17 4 nnzd ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑀 ∈ ℤ )
18 11 nnzd ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑁 ∈ ℤ )
19 prmgt1 ⊢ ( 𝑃 ∈ ℙ → 1 < 𝑃 )
20 19 ad2antrr ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ) → 1 < 𝑃 )
21 20 3adant3 ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 1 < 𝑃 )
22 simp3 ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) )
23 15 oveq1d ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 ↑ 𝑁 ) = ( 𝑄 ↑ 𝑁 ) )
24 22 23 eqtr4d ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 ↑ 𝑀 ) = ( 𝑃 ↑ 𝑁 ) )
25 16 17 18 21 24 expcand ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → 𝑀 = 𝑁 )
26 15 25 jca ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ) → ( 𝑃 = 𝑄 ∧ 𝑀 = 𝑁 ) )
27 26 3expia ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ) → ( ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) → ( 𝑃 = 𝑄 ∧ 𝑀 = 𝑁 ) ) )
28 oveq12 ⊢ ( ( 𝑃 = 𝑄 ∧ 𝑀 = 𝑁 ) → ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) )
29 27 28 impbid1 ⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ) → ( ( 𝑃 ↑ 𝑀 ) = ( 𝑄 ↑ 𝑁 ) ↔ ( 𝑃 = 𝑄 ∧ 𝑀 = 𝑁 ) ) )