Theorem rplpwr 11715

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 5782	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝐴↑𝑘) = (𝐴↑1))
2	1	oveq1d 5789	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑1) gcd 𝐵))
3	2	eqeq1d 2148	. . . . . 6 ⊢ (𝑘 = 1 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑1) gcd 𝐵) = 1))
4	3	imbi2d 229	. . . . 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 5782	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛))
6	5	oveq1d 5789	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵))
7	6	eqeq1d 2148	. . . . . 6 ⊢ (𝑘 = 𝑛 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑𝑛) gcd 𝐵) = 1))
8	7	imbi2d 229	. . . . 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 5782	. . . . . . . 8 ⊢ (𝑘 = (𝑛 + 1) → (𝐴↑𝑘) = (𝐴↑(𝑛 + 1)))
10	9	oveq1d 5789	. . . . . . 7 ⊢ (𝑘 = (𝑛 + 1) → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑(𝑛 + 1)) gcd 𝐵))
11	10	eqeq1d 2148	. . . . . 6 ⊢ (𝑘 = (𝑛 + 1) → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
12	11	imbi2d 229	. . . . 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 5782	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (𝐴↑𝑘) = (𝐴↑𝑁))
14	13	oveq1d 5789	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑁) gcd 𝐵))
15	14	eqeq1d 2148	. . . . . 6 ⊢ (𝑘 = 𝑁 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑𝑁) gcd 𝐵) = 1))
16	15	imbi2d 229	. . . . 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 8728	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
18	17	exp1d 10419	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → (𝐴↑1) = 𝐴)
19	18	oveq1d 5789	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
20	19	adantr 274	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵))
21	20	eqeq1d 2148	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝐴↑1) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
22	21	biimpar 295	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1)
23		df-3an 964	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ))
24		simpl1 984	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ)
25	24	nncnd 8734	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℂ)
26		simpl3 986	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ)
27	26	nnnn0d 9030	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ0)
28	25, 27	expp1d 10425	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = ((𝐴↑𝑛) · 𝐴))
29		simp1 981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℕ)
30		nnnn0 8984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
31	30	3ad2ant3 1004	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
32	29, 31	nnexpcld 10446	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℕ)
33	32	nnzd 9172	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℤ)
34	33	adantr 274	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℤ)
35	34	zcnd 9174	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℂ)
36	35, 25	mulcomd 7787	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) · 𝐴) = (𝐴 · (𝐴↑𝑛)))
37	28, 36	eqtrd 2172	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = (𝐴 · (𝐴↑𝑛)))
38	37	oveq2d 5790	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴 · (𝐴↑𝑛))))
39		simpl2 985	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ)
40	32	adantr 274	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℕ)
41		nnz 9073	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
42	41	3ad2ant1 1002	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℤ)
43		nnz 9073	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
44	43	3ad2ant2 1003	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℤ)
45		gcdcom 11662	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
46	42, 44, 45	syl2anc 408	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
47	46	eqeq1d 2148	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐴) = 1))
48	47	biimpa 294	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd 𝐴) = 1)
49		rpmulgcd 11714	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ (𝐴↑𝑛) ∈ ℕ) ∧ (𝐵 gcd 𝐴) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛)))
50	39, 24, 40, 48, 49	syl31anc 1219	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛)))
51	38, 50	eqtrd 2172	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴↑𝑛)))
52		peano2nn 8732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
53	52	3ad2ant3 1004	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
54	53	adantr 274	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ)
55	54	nnnn0d 9030	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ0)
56	24, 55	nnexpcld 10446	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℕ)
57	56	nnzd 9172	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℤ)
58	44	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℤ)
59		gcdcom 11662	. . . . . . . . . . . . 13 ⊢ (((𝐴↑(𝑛 + 1)) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1))))
60	57, 58, 59	syl2anc 408	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1))))
61		gcdcom 11662	. . . . . . . . . . . . 13 ⊢ (((𝐴↑𝑛) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑𝑛) gcd 𝐵) = (𝐵 gcd (𝐴↑𝑛)))
62	34, 58, 61	syl2anc 408	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = (𝐵 gcd (𝐴↑𝑛)))
63	51, 60, 62	3eqtr4d 2182	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵))
64	63	eqeq1d 2148	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1 ↔ ((𝐴↑𝑛) gcd 𝐵) = 1))
65	64	biimprd 157	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
66	23, 65	sylanbr 283	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
67	66	an32s 557	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑛 ∈ ℕ) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))
68	67	expcom 115	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)))
69	68	a2d 26	. . . . 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)))
70	4, 8, 12, 16, 22, 69	nnind 8736	. . . 4 ⊢ (𝑁 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd 𝐵) = 1))
71	70	expd 256	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1)))
72	71	com12 30	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1)))
73	72	3impia 1178	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  (class class class)co 5774  1c1 7621   + caddc 7623   · cmul 7625  ℕcn 8720  ℕ₀cn0 8977  ℤcz 9054  ↑cexp 10292   gcd cgcd 11635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636

This theorem is referenced by:  rppwr  11716