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Theorem nn0expgcd 38645
Description: Exponentiation distributes over GCD. nn0gcdsq 15946 extended to nonnegative exponents. expgcd 38644 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 11707 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 elnn0 11707 . . 3 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3 expgcd 38644 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
433expia 1101 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
5 elnn0 11707 . . . . 5 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6 0exp 13277 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0↑𝑁) = 0)
763ad2ant3 1115 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
87oveq1d 6989 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((0↑𝑁) gcd (𝐵𝑁)) = (0 gcd (𝐵𝑁)))
9 simp2 1117 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℕ)
10 nnnn0 11713 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11103ad2ant3 1115 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
129, 11nnexpcld 13419 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℕ)
1312nnzd 11897 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℤ)
14 gcd0id 15725 . . . . . . . . . 10 ((𝐵𝑁) ∈ ℤ → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1513, 14syl 17 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd (𝐵𝑁)) = (abs‘(𝐵𝑁)))
1612nnred 11454 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) ∈ ℝ)
17 0red 10441 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
1812nngt0d 11487 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵𝑁))
1917, 16, 18ltled 10586 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵𝑁))
2016, 19absidd 14641 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵𝑁)) = (𝐵𝑁))
218, 15, 203eqtrrd 2813 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = ((0↑𝑁) gcd (𝐵𝑁)))
22 oveq1 6981 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
23223ad2ant1 1113 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
24 nnz 11815 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25243ad2ant2 1114 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26 gcd0id 15725 . . . . . . . . . . 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 11445 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29 0red 10441 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 ∈ ℝ)
30 nngt0 11469 . . . . . . . . . . . . 13 (𝐵 ∈ ℕ → 0 < 𝐵)
3129, 28, 30ltled 10586 . . . . . . . . . . . 12 (𝐵 ∈ ℕ → 0 ≤ 𝐵)
3228, 31absidd 14641 . . . . . . . . . . 11 (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
33323ad2ant2 1114 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘𝐵) = 𝐵)
3423, 27, 333eqtrd 2812 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
3534oveq1d 6989 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵𝑁))
36 oveq1 6981 . . . . . . . . . 10 (𝐴 = 0 → (𝐴𝑁) = (0↑𝑁))
37363ad2ant1 1113 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
3837oveq1d 6989 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (𝐵𝑁)))
3921, 35, 383eqtr4d 2818 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
40393expia 1101 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
41 1z 11823 . . . . . . . . . 10 1 ∈ ℤ
42 gcd1 15734 . . . . . . . . . 10 (1 ∈ ℤ → (1 gcd 1) = 1)
4341, 42ax-mp 5 . . . . . . . . 9 (1 gcd 1) = 1
4443eqcomi 2781 . . . . . . . 8 1 = (1 gcd 1)
45 simp1 1116 . . . . . . . . . . . 12 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
4645oveq1d 6989 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
47 simp2 1117 . . . . . . . . . . . . 13 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℕ)
4847nnzd 11897 . . . . . . . . . . . 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 1114 . . . . . . . . . . 11 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (abs‘𝐵) = 𝐵)
5146, 49, 503eqtrd 2812 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = 𝐵)
52 simp3 1118 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝑁 = 0)
5351, 52oveq12d 6992 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
5447nncnd 11455 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
5554exp0d 13317 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
5653, 55eqtrd 2808 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
5745, 52oveq12d 6992 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
58 0exp0e1 13247 . . . . . . . . . . 11 (0↑0) = 1
5958a1i 11 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
6057, 59eqtrd 2808 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴𝑁) = 1)
6152oveq2d 6990 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = (𝐵↑0))
6261, 55eqtrd 2808 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵𝑁) = 1)
6360, 62oveq12d 6992 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = (1 gcd 1))
6444, 56, 633eqtr4a 2834 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
65643expia 1101 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
6640, 65jaod 845 . . . . 5 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
675, 66syl5bi 234 . . . 4 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
68 nnnn0 11713 . . . . . . . . . . 11 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
69683ad2ant1 1113 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℕ0)
70103ad2ant3 1115 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
7169, 70nn0expcld 13420 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) ∈ ℕ0)
72 nn0gcdid0 15727 . . . . . . . . 9 ((𝐴𝑁) ∈ ℕ0 → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
7371, 72syl 17 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd 0) = (𝐴𝑁))
74 simp2 1117 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
7574oveq1d 6989 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
7663ad2ant3 1115 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
7775, 76eqtrd 2808 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = 0)
7877oveq2d 6990 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴𝑁) gcd 0))
7974oveq2d 6990 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80 nn0gcdid0 15727 . . . . . . . . . . . 12 (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
8168, 80syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴 gcd 0) = 𝐴)
82813ad2ant1 1113 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 0) = 𝐴)
8379, 82eqtrd 2808 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
8483oveq1d 6989 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴𝑁))
8573, 78, 843eqtr4rd 2819 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
86853expia 1101 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
87 nncn 11446 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
8887exp0d 13317 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (𝐴↑0) = 1)
8988, 43syl6eqr 2826 . . . . . . . . . 10 (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
9081oveq1d 6989 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
9158a1i 11 . . . . . . . . . . 11 (𝐴 ∈ ℕ → (0↑0) = 1)
9288, 91oveq12d 6992 . . . . . . . . . 10 (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
9389, 90, 923eqtr4d 2818 . . . . . . . . 9 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
94933ad2ant1 1113 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
95 simp2 1117 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
9695oveq2d 6990 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
97 simp3 1118 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
9896, 97oveq12d 6992 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
9997oveq2d 6990 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (𝐴↑0))
10095, 97oveq12d 6992 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
10199, 100oveq12d 6992 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((𝐴↑0) gcd (0↑0)))
10294, 98, 1013eqtr4d 2818 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1031023expia 1101 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
10486, 103jaod 845 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1055, 104syl5bi 234 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
106 gcd0val 15704 . . . . . . . . . . 11 (0 gcd 0) = 0
1076, 106syl6eqr 2826 . . . . . . . . . 10 (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108106a1i 11 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109108oveq1d 6989 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
1106, 6oveq12d 6992 . . . . . . . . . 10 (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111107, 109, 1103eqtr4d 2818 . . . . . . . . 9 (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
1121113ad2ant3 1115 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
113 simp1 1116 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 = 0)
114 simp2 1117 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
115113, 114oveq12d 6992 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116115oveq1d 6989 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117113oveq1d 6989 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (0↑𝑁))
118114oveq1d 6989 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵𝑁) = (0↑𝑁))
119117, 118oveq12d 6992 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120112, 116, 1193eqtr4d 2818 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1211203expia 1101 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
12258, 43eqtr4i 2799 . . . . . . . . 9 (0↑0) = (1 gcd 1)
123106oveq1i 6984 . . . . . . . . 9 ((0 gcd 0)↑0) = (0↑0)
12458, 58oveq12i 6986 . . . . . . . . 9 ((0↑0) gcd (0↑0)) = (1 gcd 1)
125122, 123, 1243eqtr4i 2806 . . . . . . . 8 ((0 gcd 0)↑0) = ((0↑0) gcd (0↑0))
126 simp1 1116 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐴 = 0)
127 simp2 1117 . . . . . . . . . 10 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
128126, 127oveq12d 6992 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
129 simp3 1118 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
130128, 129oveq12d 6992 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131126, 129oveq12d 6992 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴𝑁) = (0↑0))
132127, 129oveq12d 6992 . . . . . . . . 9 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵𝑁) = (0↑0))
133131, 132oveq12d 6992 . . . . . . . 8 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴𝑁) gcd (𝐵𝑁)) = ((0↑0) gcd (0↑0)))
134125, 130, 1333eqtr4a 2834 . . . . . . 7 ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
1351343expia 1101 . . . . . 6 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
136121, 135jaod 845 . . . . 5 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1375, 136syl5bi 234 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1384, 67, 105, 137ccase 1018 . . 3 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1391, 2, 138syl2anb 588 . 2 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁))))
1401393impia 1097 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wo 833  w3a 1068   = wceq 1507  wcel 2050  cfv 6185  (class class class)co 6974  0cc0 10333  1c1 10334  cn 11437  0cn0 11705  cz 11791  cexp 13242  abscabs 14452   gcd cgcd 15701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-sup 8699  df-inf 8700  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-2 11501  df-3 11502  df-n0 11706  df-z 11792  df-uz 12057  df-rp 12203  df-fl 12975  df-mod 13051  df-seq 13183  df-exp 13243  df-cj 14317  df-re 14318  df-im 14319  df-sqrt 14453  df-abs 14454  df-dvds 15466  df-gcd 15702
This theorem is referenced by:  zexpgcd  38646
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