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Theorem rplpwr 16605

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 7456	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐴↑𝑘) = (𝐴↑1))
2	1	oveq1d 7463	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑1) gcd 𝐵))
3	2	eqeq1d 2742	. . . . . 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 7456	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛))
6	5	oveq1d 7463	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵))
7	6	eqeq1d 2742	. . . . . 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 7456	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝐴↑𝑘) = (𝐴↑(𝑛 + 1)))
10	9	oveq1d 7463	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑(𝑛 + 1)) gcd 𝐵))
11	10	eqeq1d 2742	. . . . . 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 7456	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (𝐴↑𝑘) = (𝐴↑𝑁))
14	13	oveq1d 7463	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑁) gcd 𝐵))
15	14	eqeq1d 2742	. . . . . 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 12301	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
18	17	exp1d 14191	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐴↑1) = 𝐴)
19	18	oveq1d 7463	. . . . . . . 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 2742	. . . . . 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 1089	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ))
24		simpl1 1191	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
25	24	nncnd 12309	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℂ)
26		simpl3 1193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ)
27	26	nnnn0d 12613	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ₀)
28	25, 27	expp1d 14197	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = ((𝐴↑𝑛) · 𝐴))
29		simp1 1136	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℕ)
30		nnnn0 12560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
31	30	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ₀)
32	29, 31	nnexpcld 14294	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℕ)
33	32	nnzd 12666	. . . . . . . . . . . . . . . . . 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 12748	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℂ)
36	35, 25	mulcomd 11311	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) · 𝐴) = (𝐴 · (𝐴↑𝑛)))
37	28, 36	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = (𝐴 · (𝐴↑𝑛)))
38	37	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴 · (𝐴↑𝑛))))
39		simpl2 1192	. . . . . . . . . . . . . 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 12660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
42	41	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
43		nnz 12660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
44	43	3ad2ant2 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℤ)
45	42, 44	gcdcomd 16560	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
46	45	eqeq1d 2742	. . . . . . . . . . . . . . 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 16604	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛)))
50	38, 49	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴↑𝑛)))
51		peano2nn 12305	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
52	51	3ad2ant3 1135	. . . . . . . . . . . . . . . . 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 12613	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ₀)
55	24, 54	nnexpcld 14294	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℕ)
56	55	nnzd 12666	. . . . . . . . . . . . 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 16560	. . . . . . . . . . . 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 16560	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = (𝐵 gcd (𝐴↑𝑛)))
60	50, 58, 59	3eqtr4d 2790	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵))
61	60	eqeq1d 2742	. . . . . . . . . 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 248	. . . . . . . . 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 651	. . . . . . 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 12311	. . . 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 1117	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 1087 = wceq 1537 ∈ wcel 2108 (class class class)co 7448 1c1 11185 + caddc 11187 · cmul 11189 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ↑cexp 14112 gcd cgcd 16540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541

This theorem is referenced by: rprpwr 16606 rppwr 16607 logbgcd1irr 26855 lgsne0 27397 2sqlem8 27488 flt4lem5a 42607 flt4lem5b 42608 flt4lem5c 42609 flt4lem5d 42610 flt4lem5e 42611

W3C validator