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Theorem rpexp 16771

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 14124	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
2	1	oveq1d 7415	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd 0) = (0 gcd 0))
3	2	eqeq1d 2767	. . . 4 ⊢ (𝑁 ∈ ℕ → (((0↑𝑁) gcd 0) = 1 ↔ (0 gcd 0) = 1))
4		oveq1 7407	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
5		oveq12 7409	. . . . . . 7 ⊢ (((𝐴↑𝑁) = (0↑𝑁) ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
6	4, 5	sylan 591	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
7	6	eqeq1d 2767	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ ((0↑𝑁) gcd 0) = 1))
8		oveq12 7409	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
9	8	eqeq1d 2767	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 ↔ (0 gcd 0) = 1))
10	7, 9	bibi12d 348	. . . 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 250	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
12	11	3ad2ant3 1151	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
13		exprmfct 16753	. . . . . . 7 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵))
14		simpl1 1208	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℤ)
15		simpl3 1210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ)
16	15	nnnn0d 12556	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ₀)
17		zexpcl 14103	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴↑𝑁) ∈ ℤ)
18	14, 16, 17	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴↑𝑁) ∈ ℤ)
19	18	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) ∈ ℤ)
20		simpl2 1209	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐵 ∈ ℤ)
21	20	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
22		gcddvds 16551	. . . . . . . . . . . . . . 15 ⊢ (((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵))
23	19, 21, 22	syl2anc 595	. . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁))
25		prmz 16723	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
26	25	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
27		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
28	14	zcnd 12692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℂ)
29		expeq0 14119	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
30	28, 15, 29	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
31	30	anbi1d 642	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) = 0 ∧ 𝐵 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
32	27, 31	mtbird 328	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0))
33		gcdn0cl 16550	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
34	18, 20, 32, 33	syl21anc 850	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
35	34	nnzd 12608	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
36	35	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
37		dvdstr 16342	. . . . . . . . . . . . . 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 1394	. . . . . . . . . . . . 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 706	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
40		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
41		simpll1 1229	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
42	15	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ)
43		prmdvdsexp 16764	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
44	40, 41, 42, 43	syl3anc 1394	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
45	39, 44	sylibd 242	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐴))
46	23	simprd 500	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵)
47		dvdstr 16342	. . . . . . . . . . . . 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 1394	. . . . . . . . . . . 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 706	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐵))
50	45, 49	jcad 521	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)))
51		dvdsgcd 16592	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
52	26, 41, 21, 51	syl3anc 1394	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
53		nprmdvds1 16755	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
54		breq2 5109	. . . . . . . . . . . . . 14 ⊢ ((𝐴 gcd 𝐵) = 1 → (𝑝 ∥ (𝐴 gcd 𝐵) ↔ 𝑝 ∥ 1))
55	54	notbid 321	. . . . . . . . . . . . 13 ⊢ ((𝐴 gcd 𝐵) = 1 → (¬ 𝑝 ∥ (𝐴 gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
56	53, 55	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → ((𝐴 gcd 𝐵) = 1 → ¬ 𝑝 ∥ (𝐴 gcd 𝐵)))
57	56	necon2ad 2975	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
58	57	adantl 486	. . . . . . . . . 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 61	. . . . . . . . 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 3166	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
61		gcdn0cl 16550	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
62	61	3adantl3 1185	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
63		eluz2b3 12937	. . . . . . . . . 10 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴 gcd 𝐵) ∈ ℕ ∧ (𝐴 gcd 𝐵) ≠ 1))
64	63	baib 544	. . . . . . . . 9 ⊢ ((𝐴 gcd 𝐵) ∈ ℕ → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ≠ 1))
65	62, 64	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ≠ 1))
66	60, 65	sylibrd 262	. . . . . . 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 35	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → (𝐴 gcd 𝐵) ∈ (ℤ_≥‘2)))
68		exprmfct 16753	. . . . . . 7 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵))
69	62	nnzd 12608	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
70	69	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∈ ℤ)
71		gcddvds 16551	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
72	41, 21, 71	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
73	72	simpld 499	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐴)
74		iddvdsexp 16327	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∥ (𝐴↑𝑁))
75	41, 42, 74	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝐴↑𝑁))
76	70, 41, 19, 73, 75	dvdstrd 16343	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁))
77		dvdstr 16342	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴 gcd 𝐵) ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁)) → 𝑝 ∥ (𝐴↑𝑁)))
78	26, 70, 19, 77	syl3anc 1394	. . . . . . . . . . . 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)))
79	76, 78	mpan2d 706	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
80	72	simprd 500	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐵)
81		dvdstr 16342	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → 𝑝 ∥ 𝐵))
82	26, 70, 21, 81	syl3anc 1394	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝐴 gcd 𝐵) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → 𝑝 ∥ 𝐵))
83	80, 82	mpan2d 706	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ 𝐵))
84	79, 83	jcad 521	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵)))
85		dvdsgcd 16592	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
86	26, 19, 21, 85	syl3anc 1394	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
87		breq2 5109	. . . . . . . . . . . . . 14 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ 𝑝 ∥ 1))
88	87	notbid 321	. . . . . . . . . . . . 13 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
89	53, 88	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → (((𝐴↑𝑁) gcd 𝐵) = 1 → ¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
90	89	necon2ad 2975	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
91	90	adantl 486	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
92	84, 86, 91	3syld 61	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
93	92	rexlimdva 3166	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
94		eluz2b3 12937	. . . . . . . . . 10 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ ∧ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
95	94	baib 544	. . . . . . . . 9 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
96	34, 95	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
97	93, 96	sylibrd 262	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2)))
98	68, 97	syl5 35	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2)))
99	67, 98	impbid 215	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ∈ (ℤ_≥‘2)))
100	99, 96, 65	3bitr3d 312	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1))
101	100	necon4bid 3005	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
102	101	ex 417	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (¬ (𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
103	12, 102	pm2.61d 181	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∃wrex 3089 class class class wbr 5105 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 ℕcn 12224 2c2 12286 ℕ₀cn0 12495 ℤcz 12582 ℤ_≥cuz 12853 ↑cexp 14088 ∥ cdvds 16300 gcd cgcd 16542 ℙcprime 16719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-n0 12496 df-z 12583 df-uz 12854 df-rp 13008 df-fz 13527 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-dvds 16301 df-gcd 16543 df-prm 16720

This theorem is referenced by: rpexp1i 16772 phiprmpw 16825 pockthlem 16955 logbgcd1irr 26917 aks4d1p8d3 42715 hashscontpow1 42750 flt4lem7 43253

W3C validator