Theorem nn0expgcd 41222

Description: Exponentiation distributes over GCD. nn0gcdsq 16685 extended to nonnegative exponents. expgcd 41221 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 12471	. . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 12471	. . 3 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		expgcd 41221	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
4	3	3expia 1122	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
5		elnn0 12471	. . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6		0exp 14060	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
7	6	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
8	7	oveq1d 7421	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((0↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd (𝐵↑𝑁)))
9		simp2 1138	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℕ)
10		nnnn0 12476	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11	10	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
12	9, 11	nnexpcld 14205	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℕ)
13	12	nnzd 12582	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℤ)
14		gcd0id 16457	. . . . . . . . . 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 12224	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℝ)
17		0red 11214	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
18	12	nngt0d 12258	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵↑𝑁))
19	17, 16, 18	ltled 11359	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵↑𝑁))
20	16, 19	absidd 15366	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵↑𝑁)) = (𝐵↑𝑁))
21	8, 15, 20	3eqtrrd 2778	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = ((0↑𝑁) gcd (𝐵↑𝑁)))
22		oveq1 7413	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
23	22	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
24		nnz 12576	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25	24	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26		gcd0id 16457	. . . . . . . . . . 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 12216	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29		0red 11214	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
30		nngt0 12240	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
31	29, 28, 30	ltled 11359	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
32	28, 31	absidd 15366	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
33	32	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘𝐵) = 𝐵)
34	23, 27, 33	3eqtrd 2777	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
35	34	oveq1d 7421	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑𝑁))
36		oveq1 7413	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
37	36	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (0↑𝑁))
38	37	oveq1d 7421	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑𝑁) gcd (𝐵↑𝑁)))
39	21, 35, 38	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
40	39	3expia 1122	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
41		1z 12589	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
42		gcd1 16466	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1)
43	41, 42	ax-mp 5	. . . . . . . . 9 ⊢ (1 gcd 1) = 1
44	43	eqcomi 2742	. . . . . . . 8 ⊢ 1 = (1 gcd 1)
45		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
46	45	oveq1d 7421	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
47		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℕ)
48	47	nnzd 12582	. . . . . . . . . . . 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 1135	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (abs‘𝐵) = 𝐵)
51	46, 49, 50	3eqtrd 2777	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = 𝐵)
52		simp3 1139	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝑁 = 0)
53	51, 52	oveq12d 7424	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
54	47	nncnd 12225	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
55	54	exp0d 14102	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
56	53, 55	eqtrd 2773	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
57	45, 52	oveq12d 7424	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴↑𝑁) = (0↑0))
58		0exp0e1 14029	. . . . . . . . . . 11 ⊢ (0↑0) = 1
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
60	57, 59	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
61	52	oveq2d 7422	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
62	61, 55	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
63	60, 62	oveq12d 7424	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
64	44, 56, 63	3eqtr4a 2799	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
65	64	3expia 1122	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
66	40, 65	jaod 858	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
67	5, 66	biimtrid 241	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
68		nnnn0 12476	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
69	68	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℕ0)
70	10	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
71	69, 70	nn0expcld 14206	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) ∈ ℕ0)
72		nn0gcdid0 16459	. . . . . . . . 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 1138	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
75	74	oveq1d 7421	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = (0↑𝑁))
76	6	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
77	75, 76	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = 0)
78	77	oveq2d 7422	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((𝐴↑𝑁) gcd 0))
79	74	oveq2d 7422	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80		nn0gcdid0 16459	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
81	68, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴 gcd 0) = 𝐴)
82	81	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 0) = 𝐴)
83	79, 82	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
84	83	oveq1d 7421	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴↑𝑁))
85	73, 78, 84	3eqtr4rd 2784	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
86	85	3expia 1122	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
87		nncn 12217	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
88	87	exp0d 14102	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑0) = 1)
89	88, 43	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
90	81	oveq1d 7421	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
91	58	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (0↑0) = 1)
92	88, 91	oveq12d 7424	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
93	89, 90, 92	3eqtr4d 2783	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
94	93	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 0)↑0) = ((𝐴↑0) gcd (0↑0)))
95		simp2 1138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
96	95	oveq2d 7422	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
97		simp3 1139	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
98	96, 97	oveq12d 7424	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
99	97	oveq2d 7422	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
100	95, 97	oveq12d 7424	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (0↑0))
101	99, 100	oveq12d 7424	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((𝐴↑0) gcd (0↑0)))
102	94, 98, 101	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
103	102	3expia 1122	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
104	86, 103	jaod 858	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
105	5, 104	biimtrid 241	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
106		gcd0val 16435	. . . . . . . . . . 11 ⊢ (0 gcd 0) = 0
107	6, 106	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108	106	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109	108	oveq1d 7421	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
110	6, 6	oveq12d 7424	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111	107, 109, 110	3eqtr4d 2783	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
112	111	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((0 gcd 0)↑𝑁) = ((0↑𝑁) gcd (0↑𝑁)))
113		simp1 1137	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐴 = 0)
114		simp2 1138	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → 𝐵 = 0)
115	113, 114	oveq12d 7424	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116	115	oveq1d 7421	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117	113	oveq1d 7421	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (0↑𝑁))
118	114	oveq1d 7421	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = (0↑𝑁))
119	117, 118	oveq12d 7424	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120	112, 116, 119	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
121	120	3expia 1122	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
122	58, 43	eqtr4i 2764	. . . . . . . . 9 ⊢ (0↑0) = (1 gcd 1)
123	106	oveq1i 7416	. . . . . . . . 9 ⊢ ((0 gcd 0)↑0) = (0↑0)
124	58, 58	oveq12i 7418	. . . . . . . . 9 ⊢ ((0↑0) gcd (0↑0)) = (1 gcd 1)
125	122, 123, 124	3eqtr4i 2771	. . . . . . . 8 ⊢ ((0 gcd 0)↑0) = ((0↑0) gcd (0↑0))
126		simp1 1137	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐴 = 0)
127		simp2 1138	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝐵 = 0)
128	126, 127	oveq12d 7424	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
129		simp3 1139	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
130	128, 129	oveq12d 7424	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131	126, 129	oveq12d 7424	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (0↑0))
132	127, 129	oveq12d 7424	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (0↑0))
133	131, 132	oveq12d 7424	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑0) gcd (0↑0)))
134	125, 130, 133	3eqtr4a 2799	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
135	134	3expia 1122	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 = 0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
136	121, 135	jaod 858	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
137	5, 136	biimtrid 241	. . . 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 599	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ0 → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))))
140	139	3impia 1118	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ‘cfv 6541  (class class class)co 7406  0cc0 11107  1c1 11108  ℕcn 12209  ℕ₀cn0 12469  ℤcz 12555  ↑cexp 14024  abscabs 15178   gcd cgcd 16432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fl 13754  df-mod 13832  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-dvds 16195  df-gcd 16433

This theorem is referenced by:  zexpgcd  41223