Theorem nn0expgcd 16524

Description: Exponentiation distributes over GCD. nn0gcdsq 16713 extended to nonnegative exponents. expgcd 16523 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 12430	. . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 12430	. . 3 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		expgcd 16523	. . . . 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 12430	. . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
6		0exp 14050	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
7	6	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0↑𝑁) = 0)
8	7	oveq1d 7375	. . . . . . . . 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 12435	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
11	10	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
12	9, 11	nnexpcld 14198	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℕ)
13	12	nnzd 12541	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℤ)
14		gcd0id 16479	. . . . . . . . . 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 12180	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) ∈ ℝ)
17		0red 11138	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ∈ ℝ)
18	12	nngt0d 12217	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 < (𝐵↑𝑁))
19	17, 16, 18	ltled 11285	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ (𝐵↑𝑁))
20	16, 19	absidd 15376	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝐵↑𝑁)) = (𝐵↑𝑁))
21	8, 15, 20	3eqtrrd 2777	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = ((0↑𝑁) gcd (𝐵↑𝑁)))
22		oveq1 7367	. . . . . . . . . . 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 12536	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
25	24	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
26		gcd0id 16479	. . . . . . . . . . 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 12172	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
29		0red 11138	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
30		nngt0 12199	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
31	29, 28, 30	ltled 11285	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
32	28, 31	absidd 15376	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐵)
35	34	oveq1d 7375	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑𝑁))
36		oveq1 7367	. . . . . . . . . 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 7375	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑𝑁) gcd (𝐵↑𝑁)))
39	21, 35, 38	3eqtr4d 2782	. . . . . . 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 12548	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
42		gcd1 16488	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1)
43	41, 42	ax-mp 5	. . . . . . . . 9 ⊢ (1 gcd 1) = 1
44	43	eqcomi 2746	. . . . . . . 8 ⊢ 1 = (1 gcd 1)
45		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐴 = 0)
46	45	oveq1d 7375	. . . . . . . . . . 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 12541	. . . . . . . . . . . 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 2776	. . . . . . . . . 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 7378	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐵↑0))
54	47	nncnd 12181	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
55	54	exp0d 14093	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑0) = 1)
56	53, 55	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = 1)
57	45, 52	oveq12d 7378	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴↑𝑁) = (0↑0))
58		0exp0e1 14019	. . . . . . . . . . 11 ⊢ (0↑0) = 1
59	58	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (0↑0) = 1)
60	57, 59	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
61	52	oveq2d 7376	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
62	61, 55	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
63	60, 62	oveq12d 7378	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
64	44, 56, 63	3eqtr4a 2798	. . . . . . 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 860	. . . . 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 12435	. . . . . . . . . . 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 14199	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) ∈ ℕ0)
72		nn0gcdid0 16481	. . . . . . . . 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 7375	. . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = 0)
78	77	oveq2d 7376	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((𝐴↑𝑁) gcd 0))
79	74	oveq2d 7376	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
80		nn0gcdid0 16481	. . . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = 𝐴)
84	83	oveq1d 7375	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = (𝐴↑𝑁))
85	73, 78, 84	3eqtr4rd 2783	. . . . . . 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 12173	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
88	87	exp0d 14093	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (𝐴↑0) = 1)
89	88, 43	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴↑0) = (1 gcd 1))
90	81	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑0) = (𝐴↑0))
91	58	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → (0↑0) = 1)
92	88, 91	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → ((𝐴↑0) gcd (0↑0)) = (1 gcd 1))
93	89, 90, 92	3eqtr4d 2782	. . . . . . . . 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 7376	. . . . . . . . 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 7378	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴 gcd 0)↑0))
99	97	oveq2d 7376	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
100	95, 97	oveq12d 7378	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (0↑0))
101	99, 100	oveq12d 7378	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((𝐴↑0) gcd (0↑0)))
102	94, 98, 101	3eqtr4d 2782	. . . . . . 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 860	. . . . 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 16457	. . . . . . . . . . 11 ⊢ (0 gcd 0) = 0
107	6, 106	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = (0 gcd 0))
108	106	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0 gcd 0) = 0)
109	108	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0 gcd 0)↑𝑁) = (0↑𝑁))
110	6, 6	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd (0↑𝑁)) = (0 gcd 0))
111	107, 109, 110	3eqtr4d 2782	. . . . . . . . 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 7378	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) = (0 gcd 0))
116	115	oveq1d 7375	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑𝑁))
117	113	oveq1d 7375	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (0↑𝑁))
118	114	oveq1d 7375	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → (𝐵↑𝑁) = (0↑𝑁))
119	117, 118	oveq12d 7378	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑𝑁) gcd (0↑𝑁)))
120	112, 116, 119	3eqtr4d 2782	. . . . . . 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 2763	. . . . . . . . 9 ⊢ (0↑0) = (1 gcd 1)
123	106	oveq1i 7370	. . . . . . . . 9 ⊢ ((0 gcd 0)↑0) = (0↑0)
124	58, 58	oveq12i 7372	. . . . . . . . 9 ⊢ ((0↑0) gcd (0↑0)) = (1 gcd 1)
125	122, 123, 124	3eqtr4i 2770	. . . . . . . 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 7378	. . . . . . . . 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 7378	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴 gcd 𝐵)↑𝑁) = ((0 gcd 0)↑0))
131	126, 129	oveq12d 7378	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (0↑0))
132	127, 129	oveq12d 7378	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (0↑0))
133	131, 132	oveq12d 7378	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = ((0↑0) gcd (0↑0)))
134	125, 130, 133	3eqtr4a 2798	. . . . . . 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 860	. . . . 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 1038	. . 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 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6492  (class class class)co 7360  0cc0 11029  1c1 11030  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ↑cexp 14014  abscabs 15187   gcd cgcd 16454

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-gcd 16455

This theorem is referenced by:  zexpgcd  16525