Theorem nn0rppwr 16618

Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵↑𝑁. rppwr 16617 extended to nonnegative integers. Less general than rpexp12i 16782. (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 12505	. 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn0 12505	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3		elnn0 12505	. . . . 5 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4		rppwr 16617	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
5	4	3expia 1137	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
6		simp1l 1214	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0)
7	6	oveq1d 7426	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (0↑𝑁))
8		0exp 14132	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
9	8	3ad2ant2 1150	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
10	7, 9	eqtrd 2804	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 0)
11	6	oveq1d 7426	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
12		simp3 1154	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13		simp1r 1215	. . . . . . . . . . . . 13 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14		nnz 12611	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15		gcd0id 16576	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
16	14, 15	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17		nnre 12239	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18		0red 11210	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
19		nngt0 12266	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
20	18, 17, 19	ltled 11357	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
21	17, 20	absidd 15473	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
22	16, 21	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = 𝐵)
23	13, 22	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0 gcd 𝐵) = 𝐵)
24	11, 12, 23	3eqtr3rd 2813	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
25	24	oveq1d 7426	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (1↑𝑁))
26		nnz 12611	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27	26	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28		1exp 14126	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
29	27, 28	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1)
30	25, 29	eqtrd 2804	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 1)
31	10, 30	oveq12d 7429	. . . . . . . 8 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd 1))
32		1z 12623	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
33		gcd0id 16576	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (0 gcd 1) = (abs‘1)
35		abs1 15347	. . . . . . . . 9 ⊢ (abs‘1) = 1
36	34, 35	eqtri 2792	. . . . . . . 8 ⊢ (0 gcd 1) = 1
37	31, 36	eqtrdi 2820	. . . . . . 7 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
38	37	3exp 1135	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
39		simp1r 1215	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0)
40	39	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
41		simp3 1154	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42		simp1l 1214	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
43	42	nnnn0d 12564	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44		nn0gcdid0 16578	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ0 → (𝐴 gcd 0) = 𝐴)
45	43, 44	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 0) = 𝐴)
46	40, 41, 45	3eqtr3rd 2813	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
47	46	oveq1d 7426	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (1↑𝑁))
48	26	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
49	48, 28	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1)
50	47, 49	eqtrd 2804	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 1)
51	39	oveq1d 7426	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (0↑𝑁))
52	8	3ad2ant2 1150	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
53	51, 52	eqtrd 2804	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 0)
54	50, 53	oveq12d 7429	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 0))
55		1nn0 12519	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
56		nn0gcdid0 16578	. . . . . . . . 9 ⊢ (1 ∈ ℕ0 → (1 gcd 0) = 1)
57	55, 56	mp1i 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1 gcd 0) = 1)
58	54, 57	eqtrd 2804	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
59	58	3exp 1135	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
60		oveq12 7420	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61		gcd0val 16554	. . . . . . . . . . . 12 ⊢ (0 gcd 0) = 0
62		0ne1 12311	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
63	61, 62	eqnetri 3034	. . . . . . . . . . 11 ⊢ (0 gcd 0) ≠ 1
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
65	60, 64	eqnetrd 3031	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
66	65	neneqd 2969	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ¬ (𝐴 gcd 𝐵) = 1)
67	66	pm2.21d 122	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
68	67	a1d 26	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
69	5, 38, 59, 68	ccase 1051	. . . . 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 609	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
71		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0))
72	71	3ad2ant3 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
73		nn0cn 12513	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
74	73	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐴 ∈ ℂ)
75	74	exp0d 14175	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑0) = 1)
76	72, 75	eqtrd 2804	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
77		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0))
78	77	3ad2ant3 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
79		nn0cn 12513	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ)
80	79	3ad2ant2 1150	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
81	80	exp0d 14175	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑0) = 1)
82	78, 81	eqtrd 2804	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
83	76, 82	oveq12d 7429	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
84		1gcd 16590	. . . . . . . 8 ⊢ (1 ∈ ℤ → (1 gcd 1) = 1)
85	32, 84	mp1i 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (1 gcd 1) = 1)
86	83, 85	eqtrd 2804	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
87	86	3expia 1137	. . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 = 0 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
88	87	a1dd 51	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 = 0 → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
89	70, 88	jaod 872	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
90	89	3impia 1133	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
91	1, 90	syl3an3b 1430	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ‘cfv 6537  (class class class)co 7411  ℂcc 11097  0cc0 11099  1c1 11100  ℕcn 12232  ℕ₀cn0 12503  ℤcz 12590  ↑cexp 14096  abscabs 15284   gcd cgcd 16551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-inf 9402  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-fl 13824  df-mod 13902  df-seq 14037  df-exp 14097  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-dvds 16310  df-gcd 16552

This theorem is referenced by:  expgcd  16620