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Theorem rplpwr 16495
Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rplpwr ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1))

Proof of Theorem rplpwr
Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7412 . . . . . . . 8 (𝑘 = 1 → (𝐴𝑘) = (𝐴↑1))
21oveq1d 7419 . . . . . . 7 (𝑘 = 1 → ((𝐴𝑘) gcd 𝐵) = ((𝐴↑1) gcd 𝐵))
32eqeq1d 2735 . . . . . 6 (𝑘 = 1 → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑1) gcd 𝐵) = 1))
43imbi2d 341 . . . . 5 (𝑘 = 1 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)))
5 oveq2 7412 . . . . . . . 8 (𝑘 = 𝑛 → (𝐴𝑘) = (𝐴𝑛))
65oveq1d 7419 . . . . . . 7 (𝑘 = 𝑛 → ((𝐴𝑘) gcd 𝐵) = ((𝐴𝑛) gcd 𝐵))
76eqeq1d 2735 . . . . . 6 (𝑘 = 𝑛 → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴𝑛) gcd 𝐵) = 1))
87imbi2d 341 . . . . 5 (𝑘 = 𝑛 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) gcd 𝐵) = 1)))
9 oveq2 7412 . . . . . . . 8 (𝑘 = (𝑛 + 1) → (𝐴𝑘) = (𝐴↑(𝑛 + 1)))
109oveq1d 7419 . . . . . . 7 (𝑘 = (𝑛 + 1) → ((𝐴𝑘) gcd 𝐵) = ((𝐴↑(𝑛 + 1)) gcd 𝐵))
1110eqeq1d 2735 . . . . . 6 (𝑘 = (𝑛 + 1) → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
1211imbi2d 341 . . . . 5 (𝑘 = (𝑛 + 1) → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
13 oveq2 7412 . . . . . . . 8 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
1413oveq1d 7419 . . . . . . 7 (𝑘 = 𝑁 → ((𝐴𝑘) gcd 𝐵) = ((𝐴𝑁) gcd 𝐵))
1514eqeq1d 2735 . . . . . 6 (𝑘 = 𝑁 → (((𝐴𝑘) gcd 𝐵) = 1 ↔ ((𝐴𝑁) gcd 𝐵) = 1))
1615imbi2d 341 . . . . 5 (𝑘 = 𝑁 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd 𝐵) = 1)))
17 nncn 12216 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
1817exp1d 14102 . . . . . . . . 9 (𝐴 ∈ ℕ → (𝐴↑1) = 𝐴)
1918oveq1d 7419 . . . . . . . 8 (𝐴 ∈ ℕ → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
2019adantr 482 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
2120eqeq1d 2735 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝐴↑1) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
2221biimpar 479 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)
23 df-3an 1090 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ))
24 simpl1 1192 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
2524nncnd 12224 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℂ)
26 simpl3 1194 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ)
2726nnnn0d 12528 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ0)
2825, 27expp1d 14108 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = ((𝐴𝑛) · 𝐴))
29 simp1 1137 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℕ)
30 nnnn0 12475 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
31303ad2ant3 1136 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
3229, 31nnexpcld 14204 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℕ)
3332nnzd 12581 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ ℤ)
3433adantr 482 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑛) ∈ ℤ)
3534zcnd 12663 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑛) ∈ ℂ)
3635, 25mulcomd 11231 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) · 𝐴) = (𝐴 · (𝐴𝑛)))
3728, 36eqtrd 2773 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = (𝐴 · (𝐴𝑛)))
3837oveq2d 7420 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴 · (𝐴𝑛))))
39 simpl2 1193 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
4032adantr 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑛) ∈ ℕ)
41 nnz 12575 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
42413ad2ant1 1134 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
43 nnz 12575 . . . . . . . . . . . . . . . . . 18 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
44433ad2ant2 1135 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℤ)
4542, 44gcdcomd 16451 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
4645eqeq1d 2735 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐴) = 1))
4746biimpa 478 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd 𝐴) = 1)
48 rpmulgcd 16494 . . . . . . . . . . . . . 14 (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ (𝐴𝑛) ∈ ℕ) ∧ (𝐵 gcd 𝐴) = 1) → (𝐵 gcd (𝐴 · (𝐴𝑛))) = (𝐵 gcd (𝐴𝑛)))
4939, 24, 40, 47, 48syl31anc 1374 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴 · (𝐴𝑛))) = (𝐵 gcd (𝐴𝑛)))
5038, 49eqtrd 2773 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴𝑛)))
51 peano2nn 12220 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
52513ad2ant3 1136 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
5352adantr 482 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ)
5453nnnn0d 12528 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ0)
5524, 54nnexpcld 14204 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℕ)
5655nnzd 12581 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℤ)
5744adantr 482 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℤ)
5856, 57gcdcomd 16451 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1))))
5934, 57gcdcomd 16451 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) gcd 𝐵) = (𝐵 gcd (𝐴𝑛)))
6050, 58, 593eqtr4d 2783 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = ((𝐴𝑛) gcd 𝐵))
6160eqeq1d 2735 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1 ↔ ((𝐴𝑛) gcd 𝐵) = 1))
6261biimprd 247 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
6323, 62sylanbr 583 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
6463an32s 651 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑛 ∈ ℕ) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
6564expcom 415 . . . . . 6 (𝑛 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
6665a2d 29 . . . . 5 (𝑛 ∈ ℕ → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑛) gcd 𝐵) = 1) → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
674, 8, 12, 16, 22, 66nnind 12226 . . . 4 (𝑁 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd 𝐵) = 1))
6867expd 417 . . 3 (𝑁 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1)))
6968com12 32 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1)))
70693impia 1118 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd 𝐵) = 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  (class class class)co 7404  1c1 11107   + caddc 11109   · cmul 11111  cn 12208  0cn0 12468  cz 12554  cexp 14023   gcd cgcd 16431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-dvds 16194  df-gcd 16432
This theorem is referenced by:  rprpwr  16496  rppwr  16497  logbgcd1irr  26279  lgsne0  26818  2sqlem8  26909  flt4lem5a  41338  flt4lem5b  41339  flt4lem5c  41340  flt4lem5d  41341  flt4lem5e  41342
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