rpexp - Intuitionistic Logic Explorer

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Theorem rpexp 10532

Description: If two numbers 𝐴 and 𝐵 are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)

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

Proof of Theorem rpexp Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0exp 9511	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
2	1	oveq1d 5547	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd 0) = (0 gcd 0))
3	2	eqeq1d 2089	. . . 4 ⊢ (𝑁 ∈ ℕ → (((0↑𝑁) gcd 0) = 1 ↔ (0 gcd 0) = 1))
4		oveq1 5539	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
5		oveq12 5541	. . . . . . 7 ⊢ (((𝐴↑𝑁) = (0↑𝑁) ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
6	4, 5	sylan 277	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
7	6	eqeq1d 2089	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ ((0↑𝑁) gcd 0) = 1))
8		oveq12 5541	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
9	8	eqeq1d 2089	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 ↔ (0 gcd 0) = 1))
10	7, 9	bibi12d 233	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1) ↔ (((0↑𝑁) gcd 0) = 1 ↔ (0 gcd 0) = 1)))
11	3, 10	syl5ibrcom 155	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
12	11	3ad2ant3 961	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
13		exprmfct 10519	. . . . . . 7 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵))
14		simpl1 941	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℤ)
15		simpl3 943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ)
16	15	nnnn0d 8341	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ₀)
17		zexpcl 9491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴↑𝑁) ∈ ℤ)
18	14, 16, 17	syl2anc 403	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴↑𝑁) ∈ ℤ)
19	18	adantr 270	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) ∈ ℤ)
20		simpl2 942	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐵 ∈ ℤ)
21	20	adantr 270	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
22		gcddvds 10355	. . . . . . . . . . . . . . 15 ⊢ (((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵))
23	19, 21, 22	syl2anc 403	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵))
24	23	simpld 110	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁))
25		prmz 10493	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
26	25	adantl 271	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
27		simpr 108	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
28	14	zcnd 8470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℂ)
29		expeq0 9507	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
30	28, 15, 29	syl2anc 403	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
31	30	anbi1d 452	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) = 0 ∧ 𝐵 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
32	27, 31	mtbird 630	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0))
33		gcdn0cl 10354	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
34	18, 20, 32, 33	syl21anc 1168	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
35	34	nnzd 8468	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
36	35	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
37		dvdstr 10232	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℤ ∧ ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ) → ((𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁)) → 𝑝 ∥ (𝐴↑𝑁)))
38	26, 36, 19, 37	syl3anc 1169	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁)) → 𝑝 ∥ (𝐴↑𝑁)))
39	24, 38	mpan2d 418	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
40		simpr 108	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
41		simpll1 977	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
42	15	adantr 270	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ)
43		prmdvdsexp 10527	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
44	40, 41, 42, 43	syl3anc 1169	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
45	39, 44	sylibd 147	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐴))
46	23	simprd 112	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵)
47		dvdstr 10232	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵) → 𝑝 ∥ 𝐵))
48	26, 36, 21, 47	syl3anc 1169	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵) → 𝑝 ∥ 𝐵))
49	46, 48	mpan2d 418	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐵))
50	45, 49	jcad 301	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)))
51		dvdsgcd 10401	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
52	26, 41, 21, 51	syl3anc 1169	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
53		nprmdvds1 10521	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
54		breq2 3789	. . . . . . . . . . . . . 14 ⊢ ((𝐴 gcd 𝐵) = 1 → (𝑝 ∥ (𝐴 gcd 𝐵) ↔ 𝑝 ∥ 1))
55	54	notbid 624	. . . . . . . . . . . . 13 ⊢ ((𝐴 gcd 𝐵) = 1 → (¬ 𝑝 ∥ (𝐴 gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
56	53, 55	syl5ibrcom 155	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → ((𝐴 gcd 𝐵) = 1 → ¬ 𝑝 ∥ (𝐴 gcd 𝐵)))
57	56	necon2ad 2302	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
58	57	adantl 271	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
59	50, 52, 58	3syld 56	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
60	59	rexlimdva 2477	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
61		gcdn0cl 10354	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
62	61	3adantl3 1096	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
63		eluz2b3 8691	. . . . . . . . . 10 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴 gcd 𝐵) ∈ ℕ ∧ (𝐴 gcd 𝐵) ≠ 1))
64	63	baib 861	. . . . . . . . 9 ⊢ ((𝐴 gcd 𝐵) ∈ ℕ → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ≠ 1))
65	62, 64	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ≠ 1))
66	60, 65	sylibrd 167	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ∈ (ℤ_≥‘2)))
67	13, 66	syl5 32	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → (𝐴 gcd 𝐵) ∈ (ℤ_≥‘2)))
68		exprmfct 10519	. . . . . . 7 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵))
69		gcddvds 10355	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
70	41, 21, 69	syl2anc 403	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
71	70	simpld 110	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐴)
72		iddvdsexp 10219	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∥ (𝐴↑𝑁))
73	41, 42, 72	syl2anc 403	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝐴↑𝑁))
74	62	nnzd 8468	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
75	74	adantr 270	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∈ ℤ)
76		dvdstr 10232	. . . . . . . . . . . . . 14 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ) → (((𝐴 gcd 𝐵) ∥ 𝐴 ∧ 𝐴 ∥ (𝐴↑𝑁)) → (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁)))
77	75, 41, 19, 76	syl3anc 1169	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (((𝐴 gcd 𝐵) ∥ 𝐴 ∧ 𝐴 ∥ (𝐴↑𝑁)) → (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁)))
78	71, 73, 77	mp2and 423	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁))
79		dvdstr 10232	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁)) → 𝑝 ∥ (𝐴↑𝑁)))
80	26, 75, 19, 79	syl3anc 1169	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁)) → 𝑝 ∥ (𝐴↑𝑁)))
81	78, 80	mpan2d 418	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
82	70	simprd 112	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐵)
83		dvdstr 10232	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → 𝑝 ∥ 𝐵))
84	26, 75, 21, 83	syl3anc 1169	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → 𝑝 ∥ 𝐵))
85	82, 84	mpan2d 418	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ 𝐵))
86	81, 85	jcad 301	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵)))
87		dvdsgcd 10401	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
88	26, 19, 21, 87	syl3anc 1169	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
89		breq2 3789	. . . . . . . . . . . . . 14 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ 𝑝 ∥ 1))
90	89	notbid 624	. . . . . . . . . . . . 13 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
91	53, 90	syl5ibrcom 155	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → (((𝐴↑𝑁) gcd 𝐵) = 1 → ¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
92	91	necon2ad 2302	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
93	92	adantl 271	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
94	86, 88, 93	3syld 56	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
95	94	rexlimdva 2477	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
96		eluz2b3 8691	. . . . . . . . . 10 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ ∧ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
97	96	baib 861	. . . . . . . . 9 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
98	34, 97	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
99	95, 98	sylibrd 167	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2)))
100	68, 99	syl5 32	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2)))
101	67, 100	impbid 127	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ∈ (ℤ_≥‘2)))
102	101, 98, 65	3bitr3d 216	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1))
103		simp1 938	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℤ)
104		simp3 940	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
105	104	nnnn0d 8341	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
106	103, 105, 17	syl2anc 403	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) ∈ ℤ)
107		simp2 939	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
108	106, 107	gcdcld 10360	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ₀)
109	108	nn0zd 8467	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
110		1zzd 8378	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℤ)
111		zdceq 8423	. . . . . . 7 ⊢ ((((𝐴↑𝑁) gcd 𝐵) ∈ ℤ ∧ 1 ∈ ℤ) → DECID ((𝐴↑𝑁) gcd 𝐵) = 1)
112	109, 110, 111	syl2anc 403	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → DECID ((𝐴↑𝑁) gcd 𝐵) = 1)
113	103, 107	gcdcld 10360	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
114	113	nn0zd 8467	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℤ)
115		zdceq 8423	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝐴 gcd 𝐵) = 1)
116	114, 110, 115	syl2anc 403	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → DECID (𝐴 gcd 𝐵) = 1)
117		nebidc 2325	. . . . . 6 ⊢ (DECID ((𝐴↑𝑁) gcd 𝐵) = 1 → (DECID (𝐴 gcd 𝐵) = 1 → ((((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1) ↔ (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1))))
118	112, 116, 117	sylc 61	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1) ↔ (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1)))
119	118	adantr 270	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1) ↔ (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1)))
120	102, 119	mpbird 165	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
121	120	ex 113	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (¬ (𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
122		gcdmndc 10340	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID (𝐴 = 0 ∧ 𝐵 = 0))
123		exmiddc 777	. . . 4 ⊢ (DECID (𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 = 0 ∧ 𝐵 = 0) ∨ ¬ (𝐴 = 0 ∧ 𝐵 = 0)))
124	122, 123	syl 14	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 = 0 ∧ 𝐵 = 0) ∨ ¬ (𝐴 = 0 ∧ 𝐵 = 0)))
125	124	3adant3 958	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 = 0 ∧ 𝐵 = 0) ∨ ¬ (𝐴 = 0 ∧ 𝐵 = 0)))
126	12, 121, 125	mpjaod 670	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 DECID wdc 775 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 ∃wrex 2349 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 ℂcc 6979 0cc0 6981 1c1 6982 ℕcn 8039 2c2 8089 ℕ₀cn0 8288 ℤcz 8351 ℤ_≥cuz 8619 ↑cexp 9475 ∥ cdvds 10195 gcd cgcd 10338 ℙcprime 10489

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-1o 6024 df-2o 6025 df-er 6129 df-en 6245 df-sup 6397 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-fzo 9153 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-dvds 10196 df-gcd 10339 df-prm 10490

This theorem is referenced by: rpexp1i 10533

W3C validator