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Theorem nn0rppwr 16598
Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 16597 extended to nonnegative integers. Less general than rpexp12i 16761. (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 12528 . 2 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 elnn0 12528 . . . . 5 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3 elnn0 12528 . . . . 5 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4 rppwr 16597 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
543expia 1122 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
6 simp1l 1198 . . . . . . . . . . 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 14138 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
983ad2ant2 1135 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
107, 9eqtrd 2777 . . . . . . . . 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 1139 . . . . . . . . . . . 12 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13 simp1r 1199 . . . . . . . . . . . . 13 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14 nnz 12634 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15 gcd0id 16556 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17 nnre 12273 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18 0red 11264 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 ∈ ℝ)
19 nngt0 12297 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 < 𝐵)
2018, 17, 19ltled 11409 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
2117, 20absidd 15461 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
2216, 21eqtrd 2777 . . . . . . . . . . . . 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 2786 . . . . . . . . . . 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 12634 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27263ad2ant2 1135 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28 1exp 14132 . . . . . . . . . . 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 2777 . . . . . . . . 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 12647 . . . . . . . . . 10 1 ∈ ℤ
33 gcd0id 16556 . . . . . . . . . 10 (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
3432, 33ax-mp 5 . . . . . . . . 9 (0 gcd 1) = (abs‘1)
35 abs1 15336 . . . . . . . . 9 (abs‘1) = 1
3634, 35eqtri 2765 . . . . . . . 8 (0 gcd 1) = 1
3731, 36eqtrdi 2793 . . . . . . 7 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
38373exp 1120 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
39 simp1r 1199 . . . . . . . . . . . . 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 1139 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42 simp1l 1198 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
4342nnnn0d 12587 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44 nn0gcdid0 16558 . . . . . . . . . . . . 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 2786 . . . . . . . . . . 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 1135 . . . . . . . . . . 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 2777 . . . . . . . . 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 1135 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
5351, 52eqtrd 2777 . . . . . . . . 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 12542 . . . . . . . . 9 1 ∈ ℕ0
56 nn0gcdid0 16558 . . . . . . . . 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 2777 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
59583exp 1120 . . . . . 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 16534 . . . . . . . . . . . 12 (0 gcd 0) = 0
62 0ne1 12337 . . . . . . . . . . . 12 0 ≠ 1
6361, 62eqnetri 3011 . . . . . . . . . . 11 (0 gcd 0) ≠ 1
6463a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
6560, 64eqnetrd 3008 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
6665neneqd 2945 . . . . . . . 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 1038 . . . . 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 1136 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
73 nn0cn 12536 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℂ)
74733ad2ant1 1134 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐴 ∈ ℂ)
7574exp0d 14180 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴↑0) = 1)
7672, 75eqtrd 2777 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = 1)
77 oveq2 7439 . . . . . . . . . 10 (𝑁 = 0 → (𝐵𝑁) = (𝐵↑0))
78773ad2ant3 1136 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
79 nn0cn 12536 . . . . . . . . . . 11 (𝐵 ∈ ℕ0𝐵 ∈ ℂ)
80793ad2ant2 1135 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐵 ∈ ℂ)
8180exp0d 14180 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵↑0) = 1)
8278, 81eqtrd 2777 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = 1)
8376, 82oveq12d 7449 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
84 1gcd 16570 . . . . . . . 8 (1 ∈ ℤ → (1 gcd 1) = 1)
8532, 84mp1i 13 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (1 gcd 1) = 1)
8683, 85eqtrd 2777 . . . . . 6 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
87863expia 1122 . . . . 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 860 . . 3 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
90893impia 1118 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
911, 90syl3an3b 1407 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cfv 6561  (class class class)co 7431  cc 11153  0cc0 11155  1c1 11156  cn 12266  0cn0 12526  cz 12613  cexp 14102  abscabs 15273   gcd cgcd 16531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-gcd 16532
This theorem is referenced by:  expgcd  16600
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