Theorem nn0rppwr 16524

Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵↑𝑁. rppwr 16523 extended to nonnegative integers. Less general than rpexp12i 16688. (Contributed by Steven Nguyen, 4-Apr-2023.)

Assertion

Ref	Expression
nn0rppwr	⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))

Proof of Theorem nn0rppwr

Step	Hyp	Ref	Expression
1		elnn0 12433	. 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn0 12433	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3		elnn0 12433	. . . . 5 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4		rppwr 16523	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
5	4	3expia 1122	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
6		simp1l 1199	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0)
7	6	oveq1d 7376	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (0↑𝑁))
8		0exp 14053	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
9	8	3ad2ant2 1135	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
10	7, 9	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 0)
11	6	oveq1d 7376	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
12		simp3 1139	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13		simp1r 1200	. . . . . . . . . . . . 13 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14		nnz 12539	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15		gcd0id 16482	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17		nnre 12175	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18		0red 11141	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
19		nngt0 12202	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
20	18, 17, 19	ltled 11288	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
21	17, 20	absidd 15379	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
22	16, 21	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = 𝐵)
23	13, 22	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0 gcd 𝐵) = 𝐵)
24	11, 12, 23	3eqtr3rd 2781	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
25	24	oveq1d 7376	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (1↑𝑁))
26		nnz 12539	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27	26	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28		1exp 14047	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1)
30	25, 29	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 1)
31	10, 30	oveq12d 7379	. . . . . . . 8 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd 1))
32		1z 12551	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
33		gcd0id 16482	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (0 gcd 1) = (abs‘1)
35		abs1 15253	. . . . . . . . 9 ⊢ (abs‘1) = 1
36	34, 35	eqtri 2760	. . . . . . . 8 ⊢ (0 gcd 1) = 1
37	31, 36	eqtrdi 2788	. . . . . . 7 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
38	37	3exp 1120	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
39		simp1r 1200	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0)
40	39	oveq2d 7377	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
41		simp3 1139	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42		simp1l 1199	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
43	42	nnnn0d 12492	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44		nn0gcdid0 16484	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 0) = 𝐴)
46	40, 41, 45	3eqtr3rd 2781	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
47	46	oveq1d 7376	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (1↑𝑁))
48	26	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
49	48, 28	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1)
50	47, 49	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 1)
51	39	oveq1d 7376	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (0↑𝑁))
52	8	3ad2ant2 1135	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
53	51, 52	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 0)
54	50, 53	oveq12d 7379	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 0))
55		1nn0 12447	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
56		nn0gcdid0 16484	. . . . . . . . 9 ⊢ (1 ∈ ℕ0 → (1 gcd 0) = 1)
57	55, 56	mp1i 13	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1 gcd 0) = 1)
58	54, 57	eqtrd 2772	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
59	58	3exp 1120	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
60		oveq12 7370	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61		gcd0val 16460	. . . . . . . . . . . 12 ⊢ (0 gcd 0) = 0
62		0ne1 12246	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
63	61, 62	eqnetri 3003	. . . . . . . . . . 11 ⊢ (0 gcd 0) ≠ 1
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
65	60, 64	eqnetrd 3000	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
66	65	neneqd 2938	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ¬ (𝐴 gcd 𝐵) = 1)
67	66	pm2.21d 121	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
68	67	a1d 25	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
69	5, 38, 59, 68	ccase 1038	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
70	2, 3, 69	syl2anb 599	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
71		oveq2 7369	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0))
72	71	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
73		nn0cn 12441	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
74	73	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐴 ∈ ℂ)
75	74	exp0d 14096	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑0) = 1)
76	72, 75	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
77		oveq2 7369	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0))
78	77	3ad2ant3 1136	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
79		nn0cn 12441	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ)
80	79	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
81	80	exp0d 14096	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑0) = 1)
82	78, 81	eqtrd 2772	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
83	76, 82	oveq12d 7379	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
84		1gcd 16496	. . . . . . . 8 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1)
85	32, 84	mp1i 13	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (1 gcd 1) = 1)
86	83, 85	eqtrd 2772	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
87	86	3expia 1122	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 = 0 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
88	87	a1dd 50	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 = 0 → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
89	70, 88	jaod 860	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
90	89	3impia 1118	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
91	1, 90	syl3an3b 1408	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6493  (class class class)co 7361  ℂcc 11030  0cc0 11032  1c1 11033  ℕcn 12168  ℕ₀cn0 12431  ℤcz 12518  ↑cexp 14017  abscabs 15190   gcd cgcd 16457

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 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

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 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-n0 12432  df-z 12519  df-uz 12783  df-rp 12937  df-fl 13745  df-mod 13823  df-seq 13958  df-exp 14018  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-dvds 16216  df-gcd 16458

This theorem is referenced by:  expgcd  16526