rpexp - Intuitionistic Logic Explorer

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Theorem rpexp 12843

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 10932	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
2	1	oveq1d 6064	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd 0) = (0 gcd 0))
3	2	eqeq1d 2241	. . . 4 ⊢ (𝑁 ∈ ℕ → (((0↑𝑁) gcd 0) = 1 ↔ (0 gcd 0) = 1))
4		oveq1 6056	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
5		oveq12 6058	. . . . . . 7 ⊢ (((𝐴↑𝑁) = (0↑𝑁) ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
6	4, 5	sylan 283	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
7	6	eqeq1d 2241	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ ((0↑𝑁) gcd 0) = 1))
8		oveq12 6058	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
9	8	eqeq1d 2241	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 ↔ (0 gcd 0) = 1))
10	7, 9	bibi12d 235	. . . 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 157	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
12	11	3ad2ant3 1047	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
13		exprmfct 12828	. . . . . . 7 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵))
14		simpl1 1027	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℤ)
15		simpl3 1029	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ)
16	15	nnnn0d 9549	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ₀)
17		zexpcl 10912	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴↑𝑁) ∈ ℤ)
18	14, 16, 17	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴↑𝑁) ∈ ℤ)
19	18	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) ∈ ℤ)
20		simpl2 1028	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐵 ∈ ℤ)
21	20	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
22		gcddvds 12652	. . . . . . . . . . . . . . 15 ⊢ (((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵))
23	19, 21, 22	syl2anc 411	. . . . . . . . . . . . . 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 112	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁))
25		prmz 12801	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
26	25	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
27		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
28	14	zcnd 9697	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℂ)
29		expeq0 10928	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
30	28, 15, 29	syl2anc 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
31	30	anbi1d 465	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) = 0 ∧ 𝐵 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
32	27, 31	mtbird 680	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0))
33		gcdn0cl 12651	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
34	18, 20, 32, 33	syl21anc 1273	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
35	34	nnzd 9695	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
36	35	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
37		dvdstr 12507	. . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . 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 428	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
40		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
41		simpll1 1063	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
42	15	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ)
43		prmdvdsexp 12838	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
44	40, 41, 42, 43	syl3anc 1274	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
45	39, 44	sylibd 149	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐴))
46	23	simprd 114	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵)
47		dvdstr 12507	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 428	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐵))
50	45, 49	jcad 307	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)))
51		dvdsgcd 12701	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
52	26, 41, 21, 51	syl3anc 1274	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
53		nprmdvds1 12830	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
54		breq2 4112	. . . . . . . . . . . . . 14 ⊢ ((𝐴 gcd 𝐵) = 1 → (𝑝 ∥ (𝐴 gcd 𝐵) ↔ 𝑝 ∥ 1))
55	54	notbid 673	. . . . . . . . . . . . 13 ⊢ ((𝐴 gcd 𝐵) = 1 → (¬ 𝑝 ∥ (𝐴 gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
56	53, 55	syl5ibrcom 157	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → ((𝐴 gcd 𝐵) = 1 → ¬ 𝑝 ∥ (𝐴 gcd 𝐵)))
57	56	necon2ad 2469	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
58	57	adantl 277	. . . . . . . . . 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 57	. . . . . . . . 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 2660	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
61		gcdn0cl 12651	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
62	61	3adantl3 1182	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
63		eluz2b3 9932	. . . . . . . . . 10 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴 gcd 𝐵) ∈ ℕ ∧ (𝐴 gcd 𝐵) ≠ 1))
64	63	baib 927	. . . . . . . . 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 169	. . . . . . 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 12828	. . . . . . 7 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵))
69		gcddvds 12652	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
70	41, 21, 69	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
71	70	simpld 112	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐴)
72		iddvdsexp 12494	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∥ (𝐴↑𝑁))
73	41, 42, 72	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝐴↑𝑁))
74	62	nnzd 9695	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
75	74	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∈ ℤ)
76		dvdstr 12507	. . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . 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 433	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁))
79		dvdstr 12507	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 428	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
82	70	simprd 114	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐵)
83		dvdstr 12507	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 428	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ 𝐵))
86	81, 85	jcad 307	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵)))
87		dvdsgcd 12701	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
88	26, 19, 21, 87	syl3anc 1274	. . . . . . . . . 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 4112	. . . . . . . . . . . . . 14 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ 𝑝 ∥ 1))
90	89	notbid 673	. . . . . . . . . . . . 13 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
91	53, 90	syl5ibrcom 157	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → (((𝐴↑𝑁) gcd 𝐵) = 1 → ¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
92	91	necon2ad 2469	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
93	92	adantl 277	. . . . . . . . . 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 57	. . . . . . . . 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 2660	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
96		eluz2b3 9932	. . . . . . . . . 10 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ ∧ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
97	96	baib 927	. . . . . . . . 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 169	. . . . . . 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 129	. . . . 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 218	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1))
103		simp1 1024	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℤ)
104		simp3 1026	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
105	104	nnnn0d 9549	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
106	103, 105, 17	syl2anc 411	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) ∈ ℤ)
107		simp2 1025	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℤ)
108	106, 107	gcdcld 12657	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ₀)
109	108	nn0zd 9694	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
110		1zzd 9600	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℤ)
111		zdceq 9649	. . . . . . 7 ⊢ ((((𝐴↑𝑁) gcd 𝐵) ∈ ℤ ∧ 1 ∈ ℤ) → DECID ((𝐴↑𝑁) gcd 𝐵) = 1)
112	109, 110, 111	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → DECID ((𝐴↑𝑁) gcd 𝐵) = 1)
113	103, 107	gcdcld 12657	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
114	113	nn0zd 9694	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℤ)
115		zdceq 9649	. . . . . . 7 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝐴 gcd 𝐵) = 1)
116	114, 110, 115	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → DECID (𝐴 gcd 𝐵) = 1)
117		nebidc 2492	. . . . . 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 62	. . . . 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 276	. . . 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 167	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
121	120	ex 115	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (¬ (𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
122		gcdmndc 12644	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID (𝐴 = 0 ∧ 𝐵 = 0))
123		exmiddc 844	. . . 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 1044	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 = 0 ∧ 𝐵 = 0) ∨ ¬ (𝐴 = 0 ∧ 𝐵 = 0)))
126	12, 121, 125	mpjaod 726	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 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 ≠ wne 2412 ∃wrex 2521 class class class wbr 4108 ‘cfv 5351 (class class class)co 6049 ℂcc 8121 0cc0 8123 1c1 8124 ℕcn 9233 2c2 9284 ℕ₀cn0 9492 ℤcz 9573 ℤ_≥cuz 9849 ↑cexp 10896 ∥ cdvds 12466 gcd cgcd 12642 ℙcprime 12797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-1o 6646 df-2o 6647 df-er 6766 df-en 6975 df-sup 7274 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-fz 10339 df-fzo 10473 df-fl 10626 df-mod 10681 df-seqfrec 10806 df-exp 10897 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677 df-dvds 12467 df-gcd 12643 df-prm 12798

This theorem is referenced by: rpexp1i 12844 phiprmpw 12912 pockthlem 13047 logbgcd1irr 15819 logbgcd1irraplemexp 15820

W3C validator