Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nn0rppwr Structured version   Visualization version   GIF version

Theorem nn0rppwr 41220
Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 16498 extended to nonnegative integers. Less general than rpexp12i 16658. (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 12471 . 2 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 elnn0 12471 . . . . 5 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3 elnn0 12471 . . . . 5 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4 rppwr 16498 . . . . . . 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 7421 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = (0↑𝑁))
8 0exp 14060 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
983ad2ant2 1135 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
107, 9eqtrd 2773 . . . . . . . . 9 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = 0)
116oveq1d 7421 . . . . . . . . . . . 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 12576 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15 gcd0id 16457 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
1614, 15syl 17 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17 nnre 12216 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18 0red 11214 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 ∈ ℝ)
19 nngt0 12240 . . . . . . . . . . . . . . . 16 (𝐵 ∈ ℕ → 0 < 𝐵)
2018, 17, 19ltled 11359 . . . . . . . . . . . . . . 15 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
2117, 20absidd 15366 . . . . . . . . . . . . . 14 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
2216, 21eqtrd 2773 . . . . . . . . . . . . 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 2782 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
2524oveq1d 7421 . . . . . . . . . 10 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = (1↑𝑁))
26 nnz 12576 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27263ad2ant2 1135 . . . . . . . . . . 11 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28 1exp 14054 . . . . . . . . . . 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 2773 . . . . . . . . 9 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = 1)
3110, 30oveq12d 7424 . . . . . . . 8 (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = (0 gcd 1))
32 1z 12589 . . . . . . . . . 10 1 ∈ ℤ
33 gcd0id 16457 . . . . . . . . . 10 (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
3432, 33ax-mp 5 . . . . . . . . 9 (0 gcd 1) = (abs‘1)
35 abs1 15241 . . . . . . . . 9 (abs‘1) = 1
3634, 35eqtri 2761 . . . . . . . 8 (0 gcd 1) = 1
3731, 36eqtrdi 2789 . . . . . . 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 7422 . . . . . . . . . . . 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 12529 . . . . . . . . . . . . 13 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44 nn0gcdid0 16459 . . . . . . . . . . . . 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 2782 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
4746oveq1d 7421 . . . . . . . . . 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 2773 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴𝑁) = 1)
5139oveq1d 7421 . . . . . . . . . 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 2773 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵𝑁) = 0)
5450, 53oveq12d 7424 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 0))
55 1nn0 12485 . . . . . . . . 9 1 ∈ ℕ0
56 nn0gcdid0 16459 . . . . . . . . 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 2773 . . . . . . 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 7415 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61 gcd0val 16435 . . . . . . . . . . . 12 (0 gcd 0) = 0
62 0ne1 12280 . . . . . . . . . . . 12 0 ≠ 1
6361, 62eqnetri 3012 . . . . . . . . . . 11 (0 gcd 0) ≠ 1
6463a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
6560, 64eqnetrd 3009 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
6665neneqd 2946 . . . . . . . 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 599 . . . 4 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1)))
71 oveq2 7414 . . . . . . . . . 10 (𝑁 = 0 → (𝐴𝑁) = (𝐴↑0))
72713ad2ant3 1136 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
73 nn0cn 12479 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℂ)
74733ad2ant1 1134 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐴 ∈ ℂ)
7574exp0d 14102 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴↑0) = 1)
7672, 75eqtrd 2773 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐴𝑁) = 1)
77 oveq2 7414 . . . . . . . . . 10 (𝑁 = 0 → (𝐵𝑁) = (𝐵↑0))
78773ad2ant3 1136 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
79 nn0cn 12479 . . . . . . . . . . 11 (𝐵 ∈ ℕ0𝐵 ∈ ℂ)
80793ad2ant2 1135 . . . . . . . . . 10 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → 𝐵 ∈ ℂ)
8180exp0d 14102 . . . . . . . . 9 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵↑0) = 1)
8278, 81eqtrd 2773 . . . . . . . 8 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (𝐵𝑁) = 1)
8376, 82oveq12d 7424 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
84 1gcd 16472 . . . . . . . 8 (1 ∈ ℤ → (1 gcd 1) = 1)
8532, 84mp1i 13 . . . . . . 7 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 = 0) → (1 gcd 1) = 1)
8683, 85eqtrd 2773 . . . . . 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 858 . . 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 1406 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  cfv 6541  (class class class)co 7406  cc 11105  0cc0 11107  1c1 11108  cn 12209  0cn0 12469  cz 12555  cexp 14024  abscabs 15178   gcd cgcd 16432
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fl 13754  df-mod 13832  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-dvds 16195  df-gcd 16433
This theorem is referenced by:  expgcd  41221
  Copyright terms: Public domain W3C validator