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Theorem nn0rppwr 16595
Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 16594 extended to nonnegative integers. Less general than rpexp12i 16758. (Contributed by Steven Nguyen, 4-Apr-2023.)
Assertion
Ref Expression
nn0rppwr ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))

Proof of Theorem nn0rppwr
StepHypRef Expression
1 elnn0 12526 . 2 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 elnn0 12526 . . . . 5 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3 elnn0 12526 . . . . 5 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4 rppwr 16594 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
543expia 1120 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
6 simp1l 1196 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0)
76oveq1d 7446 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = (0↑𝑁))
8 0exp 14135 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
983ad2ant2 1133 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
107, 9eqtrd 2775 . . . . . . . . 9 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = 0)
116oveq1d 7446 . . . . . . . . . . . 12 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
12 simp3 1137 . . . . . . . . . . . 12 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13 simp1r 1197 . . . . . . . . . . . . 13 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14 nnz 12632 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15 gcd0id 16553 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17 nnre 12271 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18 0red 11262 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 ∈ ℝ)
19 nngt0 12295 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 < 𝐵)
2018, 17, 19ltled 11407 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
2117, 20absidd 15458 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
2216, 21eqtrd 2775 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → (0 gcd 𝐵) = 𝐵)
2313, 22syl 17 . . . . . . . . . . . 12 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0 gcd 𝐵) = 𝐵)
2411, 12, 233eqtr3rd 2784 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
2524oveq1d 7446 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = (1↑𝑁))
26 nnz 12632 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27263ad2ant2 1133 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28 1exp 14129 . . . . . . . . . . 11 (𝑁 ∈ ℤ → (1↑𝑁) = 1)
2927, 28syl 17 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1)
3025, 29eqtrd 2775 . . . . . . . . 9 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = 1)
3110, 30oveq12d 7449 . . . . . . . 8 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = (0 gcd 1))
32 1z 12645 . . . . . . . . . 10 1 ∈ ℤ
33 gcd0id 16553 . . . . . . . . . 10 (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
3432, 33ax-mp 5 . . . . . . . . 9 (0 gcd 1) = (abs‘1)
35 abs1 15333 . . . . . . . . 9 (abs‘1) = 1
3634, 35eqtri 2763 . . . . . . . 8 (0 gcd 1) = 1
3731, 36eqtrdi 2791 . . . . . . 7 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
38373exp 1118 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
39 simp1r 1197 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0)
4039oveq2d 7447 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
41 simp3 1137 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42 simp1l 1196 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
4342nnnn0d 12585 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44 nn0gcdid0 16555 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
4543, 44syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 0) = 𝐴)
4640, 41, 453eqtr3rd 2784 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
4746oveq1d 7446 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = (1↑𝑁))
48263ad2ant2 1133 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
4948, 28syl 17 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1)
5047, 49eqtrd 2775 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = 1)
5139oveq1d 7446 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = (0↑𝑁))
5283ad2ant2 1133 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
5351, 52eqtrd 2775 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = 0)
5450, 53oveq12d 7449 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 0))
55 1nn0 12540 . . . . . . . . 9 1 ∈ ℕ0
56 nn0gcdid0 16555 . . . . . . . . 9 (1 ∈ ℕ0 → (1 gcd 0) = 1)
5755, 56mp1i 13 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1 gcd 0) = 1)
5854, 57eqtrd 2775 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
59583exp 1118 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
60 oveq12 7440 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61 gcd0val 16531 . . . . . . . . . . . 12 (0 gcd 0) = 0
62 0ne1 12335 . . . . . . . . . . . 12 0 ≠ 1
6361, 62eqnetri 3009 . . . . . . . . . . 11 (0 gcd 0) ≠ 1
6463a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
6560, 64eqnetrd 3006 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
6665neneqd 2943 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0) → ¬ (𝐴 gcd 𝐵) = 1)
6766pm2.21d 121 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
6867a1d 25 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
695, 38, 59, 68ccase 1037 . . . . 5 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
702, 3, 69syl2anb 598 . . . 4 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
71 oveq2 7439 . . . . . . . . . 10 (𝑁 = 0 → (𝐴𝑁) = (𝐴↑0))
72713ad2ant3 1134 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
73 nn0cn 12534 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℂ)
74733ad2ant1 1132 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐴 ∈ ℂ)
7574exp0d 14177 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴↑0) = 1)
7672, 75eqtrd 2775 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = 1)
77 oveq2 7439 . . . . . . . . . 10 (𝑁 = 0 → (𝐵𝑁) = (𝐵↑0))
78773ad2ant3 1134 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
79 nn0cn 12534 . . . . . . . . . . 11 (𝐵 ∈ ℕ0𝐵 ∈ ℂ)
80793ad2ant2 1133 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐵 ∈ ℂ)
8180exp0d 14177 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵↑0) = 1)
8278, 81eqtrd 2775 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = 1)
8376, 82oveq12d 7449 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
84 1gcd 16567 . . . . . . . 8 (1 ∈ ℤ → (1 gcd 1) = 1)
8532, 84mp1i 13 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (1 gcd 1) = 1)
8683, 85eqtrd 2775 . . . . . 6 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
87863expia 1120 . . . . 5 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 = 0 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
8887a1dd 50 . . . 4 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 = 0 → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
8970, 88jaod 859 . . 3 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
90893impia 1116 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
911, 90syl3an3b 1404 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  cc 11151  0cc0 11153  1c1 11154  cn 12264  0cn0 12524  cz 12611  cexp 14099  abscabs 15270   gcd cgcd 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529
This theorem is referenced by:  expgcd  16597
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