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Theorem nn0expgcd 40807
Description: Exponentiation distributes over GCD. nn0gcdsq 16627 extended to nonnegative exponents. expgcd 40806 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 12415 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 elnn0 12415 . . 3 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3 expgcd 40806 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
433expia 1121 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
5 elnn0 12415 . . . . 5 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6 0exp 14003 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
763ad2ant3 1135 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
87oveq1d 7372 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((0↑𝑁) gcd (𝐵𝑁)) = (0 gcd (𝐵𝑁)))
9 simp2 1137 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℕ)
10 nnnn0 12420 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11103ad2ant3 1135 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
129, 11nnexpcld 14148 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℕ)
1312nnzd 12526 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℤ)
14 gcd0id 16399 . . . . . . . . . 10 ((𝐵𝑁) ∈ ℤ → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1513, 14syl 17 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1612nnred 12168 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℝ)
17 0red 11158 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
1812nngt0d 12202 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵𝑁))
1917, 16, 18ltled 11303 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵𝑁))
2016, 19absidd 15307 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵𝑁)) = (𝐵𝑁))
218, 15, 203eqtrrd 2781 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = ((0↑𝑁) gcd (𝐵𝑁)))
22 oveq1 7364 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
23223ad2ant1 1133 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
24 nnz 12520 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25243ad2ant2 1134 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26 gcd0id 16399 . . . . . . . . . . 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 12160 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 0red 11158 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 ∈ ℝ)
30 nngt0 12184 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
3129, 28, 30ltled 11303 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
3228, 31absidd 15307 . . . . . . . . . . 11 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
33323ad2ant2 1134 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘𝐵) = 𝐵)
3423, 27, 333eqtrd 2780 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
3534oveq1d 7372 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵𝑁))
36 oveq1 7364 . . . . . . . . . 10 (𝐴 = 0 → (𝐴𝑁) = (0↑𝑁))
37363ad2ant1 1133 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
3837oveq1d 7372 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (𝐵𝑁)))
3921, 35, 383eqtr4d 2786 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
40393expia 1121 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
41 1z 12533 . . . . . . . . . 10 1 ∈ ℤ
42 gcd1 16408 . . . . . . . . . 10 (1 ∈ ℤ → (1 gcd 1) = 1)
4341, 42ax-mp 5 . . . . . . . . 9 (1 gcd 1) = 1
4443eqcomi 2745 . . . . . . . 8 1 = (1 gcd 1)
45 simp1 1136 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
4645oveq1d 7372 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
47 simp2 1137 . . . . . . . . . . . . 13 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℕ)
4847nnzd 12526 . . . . . . . . . . . 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 1134 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (abs‘𝐵) = 𝐵)
5146, 49, 503eqtrd 2780 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = 𝐵)
52 simp3 1138 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝑁 = 0)
5351, 52oveq12d 7375 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
5447nncnd 12169 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
5554exp0d 14045 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
5653, 55eqtrd 2776 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
5745, 52oveq12d 7375 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
58 0exp0e1 13972 . . . . . . . . . . 11 (0↑0) = 1
5958a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
6057, 59eqtrd 2776 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = 1)
6152oveq2d 7373 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
6261, 55eqtrd 2776 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = 1)
6360, 62oveq12d 7375 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
6444, 56, 633eqtr4a 2802 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
65643expia 1121 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
6640, 65jaod 857 . . . . 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 12420 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
69683ad2ant1 1133 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℕ0)
70103ad2ant3 1135 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
7169, 70nn0expcld 14149 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) ∈ ℕ0)
72 nn0gcdid0 16401 . . . . . . . . 9 ((𝐴𝑁) ∈ ℕ0 → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
7371, 72syl 17 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
74 simp2 1137 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
7574oveq1d 7372 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
7663ad2ant3 1135 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
7775, 76eqtrd 2776 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = 0)
7877oveq2d 7373 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴𝑁) gcd 0))
7974oveq2d 7373 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80 nn0gcdid0 16401 . . . . . . . . . . . 12 (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
8168, 80syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴 gcd 0) = 𝐴)
82813ad2ant1 1133 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 0) = 𝐴)
8379, 82eqtrd 2776 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
8483oveq1d 7372 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴𝑁))
8573, 78, 843eqtr4rd 2787 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
86853expia 1121 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
87 nncn 12161 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8887exp0d 14045 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑0) = 1)
8988, 43eqtr4di 2794 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
9081oveq1d 7372 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
9158a1i 11 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (0↑0) = 1)
9288, 91oveq12d 7375 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
9389, 90, 923eqtr4d 2786 . . . . . . . . 9 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
94933ad2ant1 1133 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
95 simp2 1137 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
9695oveq2d 7373 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
97 simp3 1138 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
9896, 97oveq12d 7375 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
9997oveq2d 7373 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
10095, 97oveq12d 7375 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
10199, 100oveq12d 7375 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴↑0) gcd (0↑0)))
10294, 98, 1013eqtr4d 2786 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1031023expia 1121 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
10486, 103jaod 857 . . . . 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 16377 . . . . . . . . . . 11 (0 gcd 0) = 0
1076, 106eqtr4di 2794 . . . . . . . . . 10 (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108106a1i 11 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109108oveq1d 7372 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
1106, 6oveq12d 7375 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111107, 109, 1103eqtr4d 2786 . . . . . . . . 9 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
1121113ad2ant3 1135 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
113 simp1 1136 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 = 0)
114 simp2 1137 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
115113, 114oveq12d 7375 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116115oveq1d 7372 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117113oveq1d 7372 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
118114oveq1d 7372 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
119117, 118oveq12d 7375 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120112, 116, 1193eqtr4d 2786 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1211203expia 1121 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
12258, 43eqtr4i 2767 . . . . . . . . 9 (0↑0) = (1 gcd 1)
123106oveq1i 7367 . . . . . . . . 9 ((0 gcd 0)↑0) = (0↑0)
12458, 58oveq12i 7369 . . . . . . . . 9 ((0↑0) gcd (0↑0)) = (1 gcd 1)
125122, 123, 1243eqtr4i 2774 . . . . . . . 8 ((0 gcd 0)↑0) = ((0↑0) gcd (0↑0))
126 simp1 1136 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐴 = 0)
127 simp2 1137 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
128126, 127oveq12d 7375 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
129 simp3 1138 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
130128, 129oveq12d 7375 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131126, 129oveq12d 7375 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
132127, 129oveq12d 7375 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
133131, 132oveq12d 7375 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑0) gcd (0↑0)))
134125, 130, 1333eqtr4a 2802 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1351343expia 1121 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
136121, 135jaod 857 . . . . 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 1036 . . 3 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1391, 2, 138syl2anb 598 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1401393impia 1117 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052  cn 12153  0cn0 12413  cz 12499  cexp 13967  abscabs 15119   gcd cgcd 16374
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375
This theorem is referenced by:  zexpgcd  40808
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