Theorem nn0rppwr 40805

Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵↑𝑁. rppwr 16440 extended to nonnegative integers. Less general than rpexp12i 16600. (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 12415	. 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn0 12415	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3		elnn0 12415	. . . . 5 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4		rppwr 16440	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
5	4	3expia 1121	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
6		simp1l 1197	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0)
7	6	oveq1d 7372	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (0↑𝑁))
8		0exp 14003	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
9	8	3ad2ant2 1134	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
10	7, 9	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 0)
11	6	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
12		simp3 1138	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13		simp1r 1198	. . . . . . . . . . . . 13 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14		nnz 12520	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15		gcd0id 16399	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17		nnre 12160	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18		0red 11158	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
19		nngt0 12184	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
20	18, 17, 19	ltled 11303	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
21	17, 20	absidd 15307	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
22	16, 21	eqtrd 2776	. . . . . . . . . . . . 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 2785	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
25	24	oveq1d 7372	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (1↑𝑁))
26		nnz 12520	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27	26	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28		1exp 13997	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 1)
31	10, 30	oveq12d 7375	. . . . . . . 8 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd 1))
32		1z 12533	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
33		gcd0id 16399	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (0 gcd 1) = (abs‘1)
35		abs1 15182	. . . . . . . . 9 ⊢ (abs‘1) = 1
36	34, 35	eqtri 2764	. . . . . . . 8 ⊢ (0 gcd 1) = 1
37	31, 36	eqtrdi 2792	. . . . . . 7 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
38	37	3exp 1119	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
39		simp1r 1198	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0)
40	39	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
41		simp3 1138	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42		simp1l 1197	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
43	42	nnnn0d 12473	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44		nn0gcdid0 16401	. . . . . . . . . . . . 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 2785	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
47	46	oveq1d 7372	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (1↑𝑁))
48	26	3ad2ant2 1134	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 1)
51	39	oveq1d 7372	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (0↑𝑁))
52	8	3ad2ant2 1134	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
53	51, 52	eqtrd 2776	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 0)
54	50, 53	oveq12d 7375	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 0))
55		1nn0 12429	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
56		nn0gcdid0 16401	. . . . . . . . 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 2776	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
59	58	3exp 1119	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
60		oveq12 7366	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61		gcd0val 16377	. . . . . . . . . . . 12 ⊢ (0 gcd 0) = 0
62		0ne1 12224	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
63	61, 62	eqnetri 3014	. . . . . . . . . . 11 ⊢ (0 gcd 0) ≠ 1
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
65	60, 64	eqnetrd 3011	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
66	65	neneqd 2948	. . . . . . . 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 1036	. . . . 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 598	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
71		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0))
72	71	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
73		nn0cn 12423	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
74	73	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐴 ∈ ℂ)
75	74	exp0d 14045	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑0) = 1)
76	72, 75	eqtrd 2776	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
77		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0))
78	77	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
79		nn0cn 12423	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ)
80	79	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
81	80	exp0d 14045	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑0) = 1)
82	78, 81	eqtrd 2776	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
83	76, 82	oveq12d 7375	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
84		1gcd 16414	. . . . . . . 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 2776	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
87	86	3expia 1121	. . . . 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 857	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
90	89	3impia 1117	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
91	1, 90	syl3an3b 1405	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  0cc0 11051  1c1 11052  ℕcn 12153  ℕ₀cn0 12413  ℤcz 12499  ↑cexp 13967  abscabs 15119   gcd cgcd 16374

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375

This theorem is referenced by:  expgcd  40806