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Theorem nn0expgcd 39422
Description: Exponentiation distributes over GCD. nn0gcdsq 16090 extended to nonnegative exponents. expgcd 39421 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 11896 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 elnn0 11896 . . 3 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3 expgcd 39421 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
433expia 1118 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
5 elnn0 11896 . . . . 5 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6 0exp 13469 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
763ad2ant3 1132 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
87oveq1d 7164 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((0↑𝑁) gcd (𝐵𝑁)) = (0 gcd (𝐵𝑁)))
9 simp2 1134 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℕ)
10 nnnn0 11901 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11103ad2ant3 1132 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
129, 11nnexpcld 13611 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℕ)
1312nnzd 12083 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℤ)
14 gcd0id 15865 . . . . . . . . . 10 ((𝐵𝑁) ∈ ℤ → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1513, 14syl 17 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1612nnred 11649 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℝ)
17 0red 10642 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
1812nngt0d 11683 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵𝑁))
1917, 16, 18ltled 10786 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵𝑁))
2016, 19absidd 14782 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵𝑁)) = (𝐵𝑁))
218, 15, 203eqtrrd 2864 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = ((0↑𝑁) gcd (𝐵𝑁)))
22 oveq1 7156 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
23223ad2ant1 1130 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
24 nnz 12001 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25243ad2ant2 1131 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26 gcd0id 15865 . . . . . . . . . . 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 11641 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 0red 10642 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 ∈ ℝ)
30 nngt0 11665 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
3129, 28, 30ltled 10786 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
3228, 31absidd 14782 . . . . . . . . . . 11 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
33323ad2ant2 1131 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘𝐵) = 𝐵)
3423, 27, 333eqtrd 2863 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
3534oveq1d 7164 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵𝑁))
36 oveq1 7156 . . . . . . . . . 10 (𝐴 = 0 → (𝐴𝑁) = (0↑𝑁))
37363ad2ant1 1130 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
3837oveq1d 7164 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (𝐵𝑁)))
3921, 35, 383eqtr4d 2869 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
40393expia 1118 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
41 1z 12009 . . . . . . . . . 10 1 ∈ ℤ
42 gcd1 15874 . . . . . . . . . 10 (1 ∈ ℤ → (1 gcd 1) = 1)
4341, 42ax-mp 5 . . . . . . . . 9 (1 gcd 1) = 1
4443eqcomi 2833 . . . . . . . 8 1 = (1 gcd 1)
45 simp1 1133 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
4645oveq1d 7164 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
47 simp2 1134 . . . . . . . . . . . . 13 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℕ)
4847nnzd 12083 . . . . . . . . . . . 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 1131 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (abs‘𝐵) = 𝐵)
5146, 49, 503eqtrd 2863 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = 𝐵)
52 simp3 1135 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝑁 = 0)
5351, 52oveq12d 7167 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
5447nncnd 11650 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
5554exp0d 13509 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
5653, 55eqtrd 2859 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
5745, 52oveq12d 7167 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
58 0exp0e1 13439 . . . . . . . . . . 11 (0↑0) = 1
5958a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
6057, 59eqtrd 2859 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = 1)
6152oveq2d 7165 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
6261, 55eqtrd 2859 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = 1)
6360, 62oveq12d 7167 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
6444, 56, 633eqtr4a 2885 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
65643expia 1118 . . . . . 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, 66syl5bi 245 . . . 4 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
68 nnnn0 11901 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
69683ad2ant1 1130 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℕ0)
70103ad2ant3 1132 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
7169, 70nn0expcld 13612 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) ∈ ℕ0)
72 nn0gcdid0 15867 . . . . . . . . 9 ((𝐴𝑁) ∈ ℕ0 → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
7371, 72syl 17 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
74 simp2 1134 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
7574oveq1d 7164 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
7663ad2ant3 1132 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
7775, 76eqtrd 2859 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = 0)
7877oveq2d 7165 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴𝑁) gcd 0))
7974oveq2d 7165 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80 nn0gcdid0 15867 . . . . . . . . . . . 12 (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
8168, 80syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴 gcd 0) = 𝐴)
82813ad2ant1 1130 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 0) = 𝐴)
8379, 82eqtrd 2859 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
8483oveq1d 7164 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴𝑁))
8573, 78, 843eqtr4rd 2870 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
86853expia 1118 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
87 nncn 11642 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8887exp0d 13509 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑0) = 1)
8988, 43syl6eqr 2877 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
9081oveq1d 7164 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
9158a1i 11 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (0↑0) = 1)
9288, 91oveq12d 7167 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
9389, 90, 923eqtr4d 2869 . . . . . . . . 9 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
94933ad2ant1 1130 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
95 simp2 1134 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
9695oveq2d 7165 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
97 simp3 1135 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
9896, 97oveq12d 7167 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
9997oveq2d 7165 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
10095, 97oveq12d 7167 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
10199, 100oveq12d 7167 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴↑0) gcd (0↑0)))
10294, 98, 1013eqtr4d 2869 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1031023expia 1118 . . . . . 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, 104syl5bi 245 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
106 gcd0val 15844 . . . . . . . . . . 11 (0 gcd 0) = 0
1076, 106syl6eqr 2877 . . . . . . . . . 10 (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108106a1i 11 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109108oveq1d 7164 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
1106, 6oveq12d 7167 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111107, 109, 1103eqtr4d 2869 . . . . . . . . 9 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
1121113ad2ant3 1132 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
113 simp1 1133 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 = 0)
114 simp2 1134 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
115113, 114oveq12d 7167 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116115oveq1d 7164 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117113oveq1d 7164 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
118114oveq1d 7164 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
119117, 118oveq12d 7167 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120112, 116, 1193eqtr4d 2869 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1211203expia 1118 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
12258, 43eqtr4i 2850 . . . . . . . . 9 (0↑0) = (1 gcd 1)
123106oveq1i 7159 . . . . . . . . 9 ((0 gcd 0)↑0) = (0↑0)
12458, 58oveq12i 7161 . . . . . . . . 9 ((0↑0) gcd (0↑0)) = (1 gcd 1)
125122, 123, 1243eqtr4i 2857 . . . . . . . 8 ((0 gcd 0)↑0) = ((0↑0) gcd (0↑0))
126 simp1 1133 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐴 = 0)
127 simp2 1134 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
128126, 127oveq12d 7167 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
129 simp3 1135 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
130128, 129oveq12d 7167 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131126, 129oveq12d 7167 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
132127, 129oveq12d 7167 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
133131, 132oveq12d 7167 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑0) gcd (0↑0)))
134125, 130, 1333eqtr4a 2885 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1351343expia 1118 . . . . . 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, 136syl5bi 245 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1384, 67, 105, 137ccase 1033 . . 3 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1391, 2, 138syl2anb 600 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1401393impia 1114 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7149  0cc0 10535  1c1 10536  cn 11634  0cn0 11894  cz 11978  cexp 13434  abscabs 14593   gcd cgcd 15841
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fl 13166  df-mod 13242  df-seq 13374  df-exp 13435  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-dvds 15608  df-gcd 15842
This theorem is referenced by:  zexpgcd  39423
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