Theorem nn0rppwr 40254

Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵↑𝑁. rppwr 16197 extended to nonnegative integers. Less general than rpexp12i 16357. (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 12165	. 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn0 12165	. . . . 5 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
3		elnn0 12165	. . . . 5 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
4		rppwr 16197	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
5	4	3expia 1119	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
6		simp1l 1195	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0)
7	6	oveq1d 7270	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (0↑𝑁))
8		0exp 13746	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
9	8	3ad2ant2 1132	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
10	7, 9	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 0)
11	6	oveq1d 7270	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
12		simp3 1136	. . . . . . . . . . . 12 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
13		simp1r 1196	. . . . . . . . . . . . 13 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
14		nnz 12272	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
15		gcd0id 16154	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
17		nnre 11910	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
18		0red 10909	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 ∈ ℝ)
19		nngt0 11934	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
20	18, 17, 19	ltled 11053	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 0 ≤ 𝐵)
21	17, 20	absidd 15062	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℕ → (abs‘𝐵) = 𝐵)
22	16, 21	eqtrd 2778	. . . . . . . . . . . . 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 2787	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1)
25	24	oveq1d 7270	. . . . . . . . . 10 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (1↑𝑁))
26		nnz 12272	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
27	26	3ad2ant2 1132	. . . . . . . . . . 11 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ)
28		1exp 13740	. . . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 1)
31	10, 30	oveq12d 7273	. . . . . . . 8 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd 1))
32		1z 12280	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
33		gcd0id 16154	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (0 gcd 1) = (abs‘1))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ (0 gcd 1) = (abs‘1)
35		abs1 14937	. . . . . . . . 9 ⊢ (abs‘1) = 1
36	34, 35	eqtri 2766	. . . . . . . 8 ⊢ (0 gcd 1) = 1
37	31, 36	eqtrdi 2795	. . . . . . 7 ⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
38	37	3exp 1117	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
39		simp1r 1196	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0)
40	39	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
41		simp3 1136	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1)
42		simp1l 1195	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
43	42	nnnn0d 12223	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ0)
44		nn0gcdid0 16156	. . . . . . . . . . . . 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 2787	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1)
47	46	oveq1d 7270	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (1↑𝑁))
48	26	3ad2ant2 1132	. . . . . . . . . . 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 2778	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 1)
51	39	oveq1d 7270	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (0↑𝑁))
52	8	3ad2ant2 1132	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0)
53	51, 52	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 0)
54	50, 53	oveq12d 7273	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 0))
55		1nn0 12179	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
56		nn0gcdid0 16156	. . . . . . . . 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 2778	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
59	58	3exp 1117	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
60		oveq12 7264	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
61		gcd0val 16132	. . . . . . . . . . . 12 ⊢ (0 gcd 0) = 0
62		0ne1 11974	. . . . . . . . . . . 12 ⊢ 0 ≠ 1
63	61, 62	eqnetri 3013	. . . . . . . . . . 11 ⊢ (0 gcd 0) ≠ 1
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠ 1)
65	60, 64	eqnetrd 3010	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1)
66	65	neneqd 2947	. . . . . . . 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 1034	. . . . 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 597	. . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
71		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0))
72	71	3ad2ant3 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = (𝐴↑0))
73		nn0cn 12173	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ)
74	73	3ad2ant1 1131	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐴 ∈ ℂ)
75	74	exp0d 13786	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑0) = 1)
76	72, 75	eqtrd 2778	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
77		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0))
78	77	3ad2ant3 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = (𝐵↑0))
79		nn0cn 12173	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ)
80	79	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → 𝐵 ∈ ℂ)
81	80	exp0d 13786	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑0) = 1)
82	78, 81	eqtrd 2778	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → (𝐵↑𝑁) = 1)
83	76, 82	oveq12d 7273	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1))
84		1gcd 16169	. . . . . . . 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 2778	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 = 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)
87	86	3expia 1119	. . . . 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 855	. . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)))
90	89	3impia 1115	. 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
91	1, 90	syl3an3b 1403	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  0cc0 10802  1c1 10803  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ↑cexp 13710  abscabs 14873   gcd cgcd 16129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-dvds 15892  df-gcd 16130

This theorem is referenced by:  expgcd  40255