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Theorem nn0rppwr 39577
 Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵↑𝑁. rppwr 15915 extended to nonnegative integers. (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 11902 . 2 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 elnn0 11902 . . . . 5 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3 elnn0 11902 . . . . 5 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4 rppwr 15915 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
543expia 1118 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
6 simp1l 1194 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0)
76oveq1d 7157 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = (0↑𝑁))
8 0exp 13477 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
983ad2ant2 1131 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
107, 9eqtrd 2833 . . . . . . . . 9 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = 0)
116oveq1d 7157 . . . . . . . . . . . 12 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
12 simp3 1135 . . . . . . . . . . . 12 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13 simp1r 1195 . . . . . . . . . . . . 13 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14 nnz 12009 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15 gcd0id 15874 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17 nnre 11647 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18 0red 10648 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 ∈ ℝ)
19 nngt0 11671 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 < 𝐵)
2018, 17, 19ltled 10792 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
2117, 20absidd 14791 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
2216, 21eqtrd 2833 . . . . . . . . . . . . 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 2842 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
2524oveq1d 7157 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = (1↑𝑁))
26 nnz 12009 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27263ad2ant2 1131 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28 1exp 13471 . . . . . . . . . . 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 2833 . . . . . . . . 9 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = 1)
3110, 30oveq12d 7160 . . . . . . . 8 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = (0 gcd 1))
32 1z 12017 . . . . . . . . . 10 1 ∈ ℤ
33 gcd0id 15874 . . . . . . . . . 10 (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
3432, 33ax-mp 5 . . . . . . . . 9 (0 gcd 1) = (abs‘1)
35 abs1 14666 . . . . . . . . 9 (abs‘1) = 1
3634, 35eqtri 2821 . . . . . . . 8 (0 gcd 1) = 1
3731, 36eqtrdi 2849 . . . . . . 7 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
38373exp 1116 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
39 simp1r 1195 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0)
4039oveq2d 7158 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
41 simp3 1135 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42 simp1l 1194 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
4342nnnn0d 11960 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44 nn0gcdid0 15876 . . . . . . . . . . . . 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 2842 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
4746oveq1d 7157 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = (1↑𝑁))
48263ad2ant2 1131 . . . . . . . . . . 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 2833 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = 1)
5139oveq1d 7157 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = (0↑𝑁))
5283ad2ant2 1131 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
5351, 52eqtrd 2833 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = 0)
5450, 53oveq12d 7160 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 0))
55 1nn0 11916 . . . . . . . . 9 1 ∈ ℕ0
56 nn0gcdid0 15876 . . . . . . . . 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 2833 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
59583exp 1116 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
60 oveq12 7151 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61 gcd0val 15853 . . . . . . . . . . . 12 (0 gcd 0) = 0
62 0ne1 11711 . . . . . . . . . . . 12 0 ≠ 1
6361, 62eqnetri 3057 . . . . . . . . . . 11 (0 gcd 0) ≠ 1
6463a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
6560, 64eqnetrd 3054 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
6665neneqd 2992 . . . . . . . 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 1033 . . . . 5 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
702, 3, 69syl2anb 600 . . . 4 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
71 oveq2 7150 . . . . . . . . . 10 (𝑁 = 0 → (𝐴𝑁) = (𝐴↑0))
72713ad2ant3 1132 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
73 nn0cn 11910 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℂ)
74733ad2ant1 1130 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐴 ∈ ℂ)
7574exp0d 13517 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴↑0) = 1)
7672, 75eqtrd 2833 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = 1)
77 oveq2 7150 . . . . . . . . . 10 (𝑁 = 0 → (𝐵𝑁) = (𝐵↑0))
78773ad2ant3 1132 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
79 nn0cn 11910 . . . . . . . . . . 11 (𝐵 ∈ ℕ0𝐵 ∈ ℂ)
80793ad2ant2 1131 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐵 ∈ ℂ)
8180exp0d 13517 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵↑0) = 1)
8278, 81eqtrd 2833 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = 1)
8376, 82oveq12d 7160 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
84 1gcd 15888 . . . . . . . 8 (1 ∈ ℤ → (1 gcd 1) = 1)
8532, 84mp1i 13 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (1 gcd 1) = 1)
8683, 85eqtrd 2833 . . . . . 6 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)
87863expia 1118 . . . . 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 856 . . 3 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
90893impia 1114 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
911, 90syl3an3b 1402 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ‘cfv 6329  (class class class)co 7142  ℂcc 10539  0cc0 10541  1c1 10542  ℕcn 11640  ℕ0cn0 11900  ℤcz 11986  ↑cexp 13442  abscabs 14602   gcd cgcd 15850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451  ax-cnex 10597  ax-resscn 10598  ax-1cn 10599  ax-icn 10600  ax-addcl 10601  ax-addrcl 10602  ax-mulcl 10603  ax-mulrcl 10604  ax-mulcom 10605  ax-addass 10606  ax-mulass 10607  ax-distr 10608  ax-i2m1 10609  ax-1ne0 10610  ax-1rid 10611  ax-rnegex 10612  ax-rrecex 10613  ax-cnre 10614  ax-pre-lttri 10615  ax-pre-lttrn 10616  ax-pre-ltadd 10617  ax-pre-mulgt0 10618  ax-pre-sup 10619 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7571  df-2nd 7682  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-sup 8905  df-inf 8906  df-pnf 10681  df-mnf 10682  df-xr 10683  df-ltxr 10684  df-le 10685  df-sub 10876  df-neg 10877  df-div 11302  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11987  df-uz 12249  df-rp 12395  df-fl 13174  df-mod 13250  df-seq 13382  df-exp 13443  df-cj 14467  df-re 14468  df-im 14469  df-sqrt 14603  df-abs 14604  df-dvds 15617  df-gcd 15851 This theorem is referenced by:  expgcd  39578
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