rpexp - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > rpexp		GIF version

Theorem rpexp 12878

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 10963	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
2	1	oveq1d 6073	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd 0) = (0 gcd 0))
3	2	eqeq1d 2243	. . . 4 ⊢ (𝑁 ∈ ℕ → (((0↑𝑁) gcd 0) = 1 ↔ (0 gcd 0) = 1))
4		oveq1 6065	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
5		oveq12 6067	. . . . . . 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 2243	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ ((0↑𝑁) gcd 0) = 1))
8		oveq12 6067	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
9	8	eqeq1d 2243	. . . . 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 12863	. . . . . . 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 9573	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ₀)
17		zexpcl 10943	. . . . . . . . . . . . . . . . 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 12687	. . . . . . . . . . . . . . 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 12836	. . . . . . . . . . . . . . 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 9722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℂ)
29		expeq0 10959	. . . . . . . . . . . . . . . . . . . 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 12686	. . . . . . . . . . . . . . . . 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 9720	. . . . . . . . . . . . . . 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 12542	. . . . . . . . . . . . . 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 12873	. . . . . . . . . . . . 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 12542	. . . . . . . . . . . . 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 12736	. . . . . . . . . . 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 12865	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
54		breq2 4118	. . . . . . . . . . . . . 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 2471	. . . . . . . . . . 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 2662	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
61		gcdn0cl 12686	. . . . . . . . . 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 9957	. . . . . . . . . 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 12863	. . . . . . 7 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵))
69		gcddvds 12687	. . . . . . . . . . . . . . 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 12529	. . . . . . . . . . . . . 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 9720	. . . . . . . . . . . . . . 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 12542	. . . . . . . . . . . . . 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 12542	. . . . . . . . . . . . 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 12542	. . . . . . . . . . . . 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 12736	. . . . . . . . . . 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 4118	. . . . . . . . . . . . . 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 2471	. . . . . . . . . . 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 2662	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
96		eluz2b3 9957	. . . . . . . . . 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 9573	. . . . . . . . . 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 12692	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ₀)
109	108	nn0zd 9719	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
110		1zzd 9624	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℤ)
111		zdceq 9673	. . . . . . 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 12692	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ₀)
114	113	nn0zd 9719	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℤ)
115		zdceq 9673	. . . . . . 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 2494	. . . . . 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 12679	. . . 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 2205 ≠ wne 2414 ∃wrex 2523 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 0cc0 8143 1c1 8144 ℕcn 9257 2c2 9308 ℕ₀cn0 9516 ℤcz 9597 ℤ_≥cuz 9874 ↑cexp 10927 ∥ cdvds 12501 gcd cgcd 12677 ℙcprime 12832

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-2o 6661 df-er 6780 df-en 6989 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-fzo 10502 df-fl 10657 df-mod 10712 df-seqfrec 10837 df-exp 10928 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712 df-dvds 12502 df-gcd 12678 df-prm 12833

This theorem is referenced by: rpexp1i 12879 phiprmpw 12947 pockthlem 13082 logbgcd1irr 15961 logbgcd1irraplemexp 15962

W3C validator