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Theorem rplpwr 16504

Description: If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
rplpwr	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1))

Proof of Theorem rplpwr Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7412	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐴↑𝑘) = (𝐴↑1))
2	1	oveq1d 7419	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑1) gcd 𝐵))
3	2	eqeq1d 2728	. . . . . 6 ⊢ (𝑘 = 1 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑1) gcd 𝐵) = 1))
4	3	imbi2d 340	. . . . 5 ⊢ (𝑘 = 1 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)))
5		oveq2 7412	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛))
6	5	oveq1d 7419	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵))
7	6	eqeq1d 2728	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑𝑛) gcd 𝐵) = 1))
8	7	imbi2d 340	. . . . 5 ⊢ (𝑘 = 𝑛 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = 1)))
9		oveq2 7412	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝐴↑𝑘) = (𝐴↑(𝑛 + 1)))
10	9	oveq1d 7419	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑(𝑛 + 1)) gcd 𝐵))
11	10	eqeq1d 2728	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
12	11	imbi2d 340	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
13		oveq2 7412	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (𝐴↑𝑘) = (𝐴↑𝑁))
14	13	oveq1d 7419	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑁) gcd 𝐵))
15	14	eqeq1d 2728	. . . . . 6 ⊢ (𝑘 = 𝑁 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑𝑁) gcd 𝐵) = 1))
16	15	imbi2d 340	. . . . 5 ⊢ (𝑘 = 𝑁 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd 𝐵) = 1)))
17		nncn 12221	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
18	17	exp1d 14109	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐴↑1) = 𝐴)
19	18	oveq1d 7419	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
20	19	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
21	20	eqeq1d 2728	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝐴↑1) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
22	21	biimpar 477	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)
23		df-3an 1086	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ))
24		simpl1 1188	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
25	24	nncnd 12229	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℂ)
26		simpl3 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ)
27	26	nnnn0d 12533	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ₀)
28	25, 27	expp1d 14115	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = ((𝐴↑𝑛) · 𝐴))
29		simp1 1133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℕ)
30		nnnn0 12480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
31	30	3ad2ant3 1132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
32	29, 31	nnexpcld 14211	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℕ)
33	32	nnzd 12586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℤ)
34	33	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℤ)
35	34	zcnd 12668	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℂ)
36	35, 25	mulcomd 11236	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) · 𝐴) = (𝐴 · (𝐴↑𝑛)))
37	28, 36	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = (𝐴 · (𝐴↑𝑛)))
38	37	oveq2d 7420	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴 · (𝐴↑𝑛))))
39		simpl2 1189	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
40	32	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℕ)
41		nnz 12580	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
42	41	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
43		nnz 12580	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
44	43	3ad2ant2 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℤ)
45	42, 44	gcdcomd 16460	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
46	45	eqeq1d 2728	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐴) = 1))
47	46	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd 𝐴) = 1)
48		rpmulgcd 16503	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ (𝐴↑𝑛) ∈ ℕ) ∧ (𝐵 gcd 𝐴) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛)))
49	39, 24, 40, 47, 48	syl31anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛)))
50	38, 49	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴↑𝑛)))
51		peano2nn 12225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
52	51	3ad2ant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
53	52	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ)
54	53	nnnn0d 12533	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ₀)
55	24, 54	nnexpcld 14211	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℕ)
56	55	nnzd 12586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℤ)
57	44	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℤ)
58	56, 57	gcdcomd 16460	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1))))
59	34, 57	gcdcomd 16460	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = (𝐵 gcd (𝐴↑𝑛)))
60	50, 58, 59	3eqtr4d 2776	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵))
61	60	eqeq1d 2728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1 ↔ ((𝐴↑𝑛) gcd 𝐵) = 1))
62	61	biimprd 247	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
63	23, 62	sylanbr 581	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
64	63	an32s 649	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑛 ∈ ℕ) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
65	64	expcom 413	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
66	65	a2d 29	. . . . 5 ⊢ (𝑛 ∈ ℕ → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = 1) → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
67	4, 8, 12, 16, 22, 66	nnind 12231	. . . 4 ⊢ (𝑁 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd 𝐵) = 1))
68	67	expd 415	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1)))
69	68	com12 32	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1)))
70	69	3impia 1114	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 (class class class)co 7404 1c1 11110 + caddc 11112 · cmul 11114 ℕcn 12213 ℕ₀cn0 12473 ℤcz 12559 ↑cexp 14030 gcd cgcd 16440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16203 df-gcd 16441

This theorem is referenced by: rprpwr 16505 rppwr 16506 logbgcd1irr 26677 lgsne0 27219 2sqlem8 27310 flt4lem5a 41953 flt4lem5b 41954 flt4lem5c 41955 flt4lem5d 41956 flt4lem5e 41957

W3C validator