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Theorem nn0expgcd 41691
Description: Exponentiation distributes over GCD. nn0gcdsq 16695 extended to nonnegative exponents. expgcd 41690 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
Assertion
Ref Expression
nn0expgcd ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))

Proof of Theorem nn0expgcd
StepHypRef Expression
1 elnn0 12481 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 elnn0 12481 . . 3 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3 expgcd 41690 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
433expia 1120 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
5 elnn0 12481 . . . . 5 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6 0exp 14070 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
763ad2ant3 1134 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
87oveq1d 7427 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((0↑𝑁) gcd (𝐵𝑁)) = (0 gcd (𝐵𝑁)))
9 simp2 1136 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℕ)
10 nnnn0 12486 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11103ad2ant3 1134 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
129, 11nnexpcld 14215 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℕ)
1312nnzd 12592 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℤ)
14 gcd0id 16467 . . . . . . . . . 10 ((𝐵𝑁) ∈ ℤ → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1513, 14syl 17 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1612nnred 12234 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℝ)
17 0red 11224 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
1812nngt0d 12268 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵𝑁))
1917, 16, 18ltled 11369 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵𝑁))
2016, 19absidd 15376 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵𝑁)) = (𝐵𝑁))
218, 15, 203eqtrrd 2776 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = ((0↑𝑁) gcd (𝐵𝑁)))
22 oveq1 7419 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
23223ad2ant1 1132 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
24 nnz 12586 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25243ad2ant2 1133 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26 gcd0id 16467 . . . . . . . . . . 11 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
2725, 26syl 17 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd 𝐵) = (abs‘𝐵))
28 nnre 12226 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 0red 11224 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 ∈ ℝ)
30 nngt0 12250 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
3129, 28, 30ltled 11369 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
3228, 31absidd 15376 . . . . . . . . . . 11 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
33323ad2ant2 1133 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘𝐵) = 𝐵)
3423, 27, 333eqtrd 2775 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
3534oveq1d 7427 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵𝑁))
36 oveq1 7419 . . . . . . . . . 10 (𝐴 = 0 → (𝐴𝑁) = (0↑𝑁))
37363ad2ant1 1132 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
3837oveq1d 7427 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (𝐵𝑁)))
3921, 35, 383eqtr4d 2781 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
40393expia 1120 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
41 1z 12599 . . . . . . . . . 10 1 ∈ ℤ
42 gcd1 16476 . . . . . . . . . 10 (1 ∈ ℤ → (1 gcd 1) = 1)
4341, 42ax-mp 5 . . . . . . . . 9 (1 gcd 1) = 1
4443eqcomi 2740 . . . . . . . 8 1 = (1 gcd 1)
45 simp1 1135 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
4645oveq1d 7427 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
47 simp2 1136 . . . . . . . . . . . . 13 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℕ)
4847nnzd 12592 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℤ)
4948, 26syl 17 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0 gcd 𝐵) = (abs‘𝐵))
50323ad2ant2 1133 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (abs‘𝐵) = 𝐵)
5146, 49, 503eqtrd 2775 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = 𝐵)
52 simp3 1137 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝑁 = 0)
5351, 52oveq12d 7430 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
5447nncnd 12235 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
5554exp0d 14112 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
5653, 55eqtrd 2771 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
5745, 52oveq12d 7430 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
58 0exp0e1 14039 . . . . . . . . . . 11 (0↑0) = 1
5958a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
6057, 59eqtrd 2771 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = 1)
6152oveq2d 7428 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
6261, 55eqtrd 2771 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = 1)
6360, 62oveq12d 7430 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
6444, 56, 633eqtr4a 2797 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
65643expia 1120 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
6640, 65jaod 856 . . . . 5 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
675, 66biimtrid 241 . . . 4 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
68 nnnn0 12486 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
69683ad2ant1 1132 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℕ0)
70103ad2ant3 1134 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
7169, 70nn0expcld 14216 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) ∈ ℕ0)
72 nn0gcdid0 16469 . . . . . . . . 9 ((𝐴𝑁) ∈ ℕ0 → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
7371, 72syl 17 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
74 simp2 1136 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
7574oveq1d 7427 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
7663ad2ant3 1134 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
7775, 76eqtrd 2771 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = 0)
7877oveq2d 7428 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴𝑁) gcd 0))
7974oveq2d 7428 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80 nn0gcdid0 16469 . . . . . . . . . . . 12 (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
8168, 80syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴 gcd 0) = 𝐴)
82813ad2ant1 1132 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 0) = 𝐴)
8379, 82eqtrd 2771 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
8483oveq1d 7427 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴𝑁))
8573, 78, 843eqtr4rd 2782 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
86853expia 1120 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
87 nncn 12227 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8887exp0d 14112 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑0) = 1)
8988, 43eqtr4di 2789 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
9081oveq1d 7427 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
9158a1i 11 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (0↑0) = 1)
9288, 91oveq12d 7430 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
9389, 90, 923eqtr4d 2781 . . . . . . . . 9 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
94933ad2ant1 1132 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
95 simp2 1136 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
9695oveq2d 7428 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
97 simp3 1137 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
9896, 97oveq12d 7430 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
9997oveq2d 7428 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
10095, 97oveq12d 7430 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
10199, 100oveq12d 7430 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴↑0) gcd (0↑0)))
10294, 98, 1013eqtr4d 2781 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1031023expia 1120 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
10486, 103jaod 856 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1055, 104biimtrid 241 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
106 gcd0val 16445 . . . . . . . . . . 11 (0 gcd 0) = 0
1076, 106eqtr4di 2789 . . . . . . . . . 10 (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108106a1i 11 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109108oveq1d 7427 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
1106, 6oveq12d 7430 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111107, 109, 1103eqtr4d 2781 . . . . . . . . 9 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
1121113ad2ant3 1134 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
113 simp1 1135 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 = 0)
114 simp2 1136 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
115113, 114oveq12d 7430 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116115oveq1d 7427 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117113oveq1d 7427 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
118114oveq1d 7427 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
119117, 118oveq12d 7430 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120112, 116, 1193eqtr4d 2781 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1211203expia 1120 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
12258, 43eqtr4i 2762 . . . . . . . . 9 (0↑0) = (1 gcd 1)
123106oveq1i 7422 . . . . . . . . 9 ((0 gcd 0)↑0) = (0↑0)
12458, 58oveq12i 7424 . . . . . . . . 9 ((0↑0) gcd (0↑0)) = (1 gcd 1)
125122, 123, 1243eqtr4i 2769 . . . . . . . 8 ((0 gcd 0)↑0) = ((0↑0) gcd (0↑0))
126 simp1 1135 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐴 = 0)
127 simp2 1136 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
128126, 127oveq12d 7430 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
129 simp3 1137 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
130128, 129oveq12d 7430 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131126, 129oveq12d 7430 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
132127, 129oveq12d 7430 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
133131, 132oveq12d 7430 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑0) gcd (0↑0)))
134125, 130, 1333eqtr4a 2797 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1351343expia 1120 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
136121, 135jaod 856 . . . . 5 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1375, 136biimtrid 241 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1384, 67, 105, 137ccase 1035 . . 3 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1391, 2, 138syl2anb 597 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1401393impia 1116 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  cfv 6543  (class class class)co 7412  0cc0 11116  1c1 11117  cn 12219  0cn0 12479  cz 12565  cexp 14034  abscabs 15188   gcd cgcd 16442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-fl 13764  df-mod 13842  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-dvds 16205  df-gcd 16443
This theorem is referenced by:  zexpgcd  41692
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