Theorem nn0expgcd 16583

Description: Exponentiation distributes over GCD. nn0gcdsq 16771 extended to nonnegative exponents. expgcd 16582 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)

Assertion

Ref	Expression
nn0expgcd	⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))

Proof of Theorem nn0expgcd

Step	Hyp	Ref	Expression
1		elnn0 12511	. . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 12511	. . 3 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		expgcd 16582	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
4	3	3expia 1121	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
5		elnn0 12511	. . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6		0exp 14120	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
7	6	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
8	7	oveq1d 7428	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((0↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd (𝐵↑𝑁)))
9		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℕ)
10		nnnn0 12516	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11	10	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
12	9, 11	nnexpcld 14266	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℕ)
13	12	nnzd 12623	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℤ)
14		gcd0id 16538	. . . . . . . . . 10 ⊢ ((𝐵↑𝑁) ∈ ℤ → (0 gcd (𝐵↑𝑁)) = (abs‘(𝐵↑𝑁)))
15	13, 14	syl 17	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd (𝐵↑𝑁)) = (abs‘(𝐵↑𝑁)))
16	12	nnred 12263	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℝ)
17		0red 11246	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
18	12	nngt0d 12297	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵↑𝑁))
19	17, 16, 18	ltled 11391	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵↑𝑁))
20	16, 19	absidd 15443	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵↑𝑁)) = (𝐵↑𝑁))
21	8, 15, 20	3eqtrrd 2774	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = ((0↑𝑁) gcd (𝐵↑𝑁)))
22		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
23	22	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
24		nnz 12617	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25	24	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26		gcd0id 16538	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 gcd 𝐵) = (abs‘𝐵))
28		nnre 12255	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29		0red 11246	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
30		nngt0 12279	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
31	29, 28, 30	ltled 11391	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
32	28, 31	absidd 15443	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
33	32	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘𝐵) = 𝐵)
34	23, 27, 33	3eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
35	34	oveq1d 7428	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑𝑁))
36		oveq1 7420	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
37	36	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (0↑𝑁))
38	37	oveq1d 7428	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑𝑁) gcd (𝐵↑𝑁)))
39	21, 35, 38	3eqtr4d 2779	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
40	39	3expia 1121	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
41		1z 12630	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
42		gcd1 16547	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1)
43	41, 42	ax-mp 5	. . . . . . . . 9 ⊢ (1 gcd 1) = 1
44	43	eqcomi 2743	. . . . . . . 8 ⊢ 1 = (1 gcd 1)
45		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
46	45	oveq1d 7428	. . . . . . . . . . 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)
48	47	nnzd 12623	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℤ)
49	48, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0 gcd 𝐵) = (abs‘𝐵))
50	32	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (abs‘𝐵) = 𝐵)
51	46, 49, 50	3eqtrd 2773	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = 𝐵)
52		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝑁 = 0)
53	51, 52	oveq12d 7431	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
54	47	nncnd 12264	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
55	54	exp0d 14162	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
56	53, 55	eqtrd 2769	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
57	45, 52	oveq12d 7431	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴↑𝑁) = (0↑0))
58		0exp0e1 14089	. . . . . . . . . . 11 ⊢ (0↑0) = 1
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
60	57, 59	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
61	52	oveq2d 7429	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
62	61, 55	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
63	60, 62	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
64	44, 56, 63	3eqtr4a 2795	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
65	64	3expia 1121	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
66	40, 65	jaod 859	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
67	5, 66	biimtrid 242	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
68		nnnn0 12516	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
69	68	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℕ0)
70	10	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
71	69, 70	nn0expcld 14267	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) ∈ ℕ0)
72		nn0gcdid0 16540	. . . . . . . . 9 ⊢ ((𝐴↑𝑁) ∈ ℕ0 → ((𝐴↑𝑁) gcd 0) = (𝐴↑𝑁))
73	71, 72	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 0) = (𝐴↑𝑁))
74		simp2 1137	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
75	74	oveq1d 7428	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = (0↑𝑁))
76	6	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
77	75, 76	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = 0)
78	77	oveq2d 7429	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((𝐴↑𝑁) gcd 0))
79	74	oveq2d 7429	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80		nn0gcdid0 16540	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
81	68, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴 gcd 0) = 𝐴)
82	81	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 0) = 𝐴)
83	79, 82	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
84	83	oveq1d 7428	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴↑𝑁))
85	73, 78, 84	3eqtr4rd 2780	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
86	85	3expia 1121	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
87		nncn 12256	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
88	87	exp0d 14162	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑0) = 1)
89	88, 43	eqtr4di 2787	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
90	81	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
91	58	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (0↑0) = 1)
92	88, 91	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
93	89, 90, 92	3eqtr4d 2779	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
94	93	3ad2ant1 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)
96	95	oveq2d 7429	. . . . . . . . 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)
98	96, 97	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
99	97	oveq2d 7429	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
100	95, 97	oveq12d 7431	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (0↑0))
101	99, 100	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((𝐴↑0) gcd (0↑0)))
102	94, 98, 101	3eqtr4d 2779	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
103	102	3expia 1121	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
104	86, 103	jaod 859	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
105	5, 104	biimtrid 242	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
106		gcd0val 16516	. . . . . . . . . . 11 ⊢ (0 gcd 0) = 0
107	6, 106	eqtr4di 2787	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108	106	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109	108	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
110	6, 6	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111	107, 109, 110	3eqtr4d 2779	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
112	111	3ad2ant3 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)
115	113, 114	oveq12d 7431	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116	115	oveq1d 7428	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117	113	oveq1d 7428	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (0↑𝑁))
118	114	oveq1d 7428	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = (0↑𝑁))
119	117, 118	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120	112, 116, 119	3eqtr4d 2779	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
121	120	3expia 1121	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
122	58, 43	eqtr4i 2760	. . . . . . . . 9 ⊢ (0↑0) = (1 gcd 1)
123	106	oveq1i 7423	. . . . . . . . 9 ⊢ ((0 gcd 0)↑0) = (0↑0)
124	58, 58	oveq12i 7425	. . . . . . . . 9 ⊢ ((0↑0) gcd (0↑0)) = (1 gcd 1)
125	122, 123, 124	3eqtr4i 2767	. . . . . . . 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)
128	126, 127	oveq12d 7431	. . . . . . . . 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)
130	128, 129	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131	126, 129	oveq12d 7431	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (0↑0))
132	127, 129	oveq12d 7431	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (0↑0))
133	131, 132	oveq12d 7431	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑0) gcd (0↑0)))
134	125, 130, 133	3eqtr4a 2795	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
135	134	3expia 1121	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
136	121, 135	jaod 859	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
137	5, 136	biimtrid 242	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
138	4, 67, 105, 137	ccase 1037	. . 3 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
139	1, 2, 138	syl2anb 598	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
140	139	3impia 1117	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ‘cfv 6541  (class class class)co 7413  0cc0 11137  1c1 11138  ℕcn 12248  ℕ₀cn0 12509  ℤcz 12596  ↑cexp 14084  abscabs 15255   gcd cgcd 16513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-sup 9464  df-inf 9465  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-2 12311  df-3 12312  df-n0 12510  df-z 12597  df-uz 12861  df-rp 13017  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-cj 15120  df-re 15121  df-im 15122  df-sqrt 15256  df-abs 15257  df-dvds 16273  df-gcd 16514

This theorem is referenced by:  zexpgcd  16584