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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  ℕcn 11634  ℕ0cn0 11894  ℤcz 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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