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Theorem rpexp 16746

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 14120	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
2	1	oveq1d 7425	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((0↑𝑁) gcd 0) = (0 gcd 0))
3	2	eqeq1d 2738	. . . 4 ⊢ (𝑁 ∈ ℕ → (((0↑𝑁) gcd 0) = 1 ↔ (0 gcd 0) = 1))
4		oveq1 7417	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁))
5		oveq12 7419	. . . . . . 7 ⊢ (((𝐴↑𝑁) = (0↑𝑁) ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
6	4, 5	sylan 580	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑𝑁) gcd 𝐵) = ((0↑𝑁) gcd 0))
7	6	eqeq1d 2738	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ ((0↑𝑁) gcd 0) = 1))
8		oveq12 7419	. . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
9	8	eqeq1d 2738	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 ↔ (0 gcd 0) = 1))
10	7, 9	bibi12d 345	. . . 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 247	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
12	11	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 = 0 ∧ 𝐵 = 0) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)))
13		exprmfct 16728	. . . . . . 7 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵))
14		simpl1 1192	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℤ)
15		simpl3 1194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ)
16	15	nnnn0d 12567	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝑁 ∈ ℕ₀)
17		zexpcl 14099	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝐴↑𝑁) ∈ ℤ)
18	14, 16, 17	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴↑𝑁) ∈ ℤ)
19	18	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴↑𝑁) ∈ ℤ)
20		simpl2 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐵 ∈ ℤ)
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
22		gcddvds 16527	. . . . . . . . . . . . . . 15 ⊢ (((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁) ∧ ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵))
23	19, 21, 22	syl2anc 584	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ (𝐴↑𝑁))
25		prmz 16699	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
26	25	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
27		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
28	14	zcnd 12703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → 𝐴 ∈ ℂ)
29		expeq0 14115	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
30	28, 15, 29	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
31	30	anbi1d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) = 0 ∧ 𝐵 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
32	27, 31	mtbird 325	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0))
33		gcdn0cl 16526	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ ((𝐴↑𝑁) = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
34	18, 20, 32, 33	syl21anc 837	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℕ)
35	34	nnzd 12620	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
36	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∈ ℤ)
37		dvdstr 16318	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 694	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
40		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
41		simpll1 1213	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
42	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝑁 ∈ ℕ)
43		prmdvdsexp 16739	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
44	40, 41, 42, 43	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴↑𝑁) ↔ 𝑝 ∥ 𝐴))
45	39, 44	sylibd 239	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐴))
46	23	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴↑𝑁) gcd 𝐵) ∥ 𝐵)
47		dvdstr 16318	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 694	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → 𝑝 ∥ 𝐵))
50	45, 49	jcad 512	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵)))
51		dvdsgcd 16568	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
52	26, 41, 21, 51	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ (𝐴 gcd 𝐵)))
53		nprmdvds1 16730	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → ¬ 𝑝 ∥ 1)
54		breq2 5128	. . . . . . . . . . . . . 14 ⊢ ((𝐴 gcd 𝐵) = 1 → (𝑝 ∥ (𝐴 gcd 𝐵) ↔ 𝑝 ∥ 1))
55	54	notbid 318	. . . . . . . . . . . . 13 ⊢ ((𝐴 gcd 𝐵) = 1 → (¬ 𝑝 ∥ (𝐴 gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
56	53, 55	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → ((𝐴 gcd 𝐵) = 1 → ¬ 𝑝 ∥ (𝐴 gcd 𝐵)))
57	56	necon2ad 2948	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
58	57	adantl 481	. . . . . . . . . 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 60	. . . . . . . . 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 3142	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → (𝐴 gcd 𝐵) ≠ 1))
61		gcdn0cl 16526	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
62	61	3adantl3 1169	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
63		eluz2b3 12943	. . . . . . . . . 10 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴 gcd 𝐵) ∈ ℕ ∧ (𝐴 gcd 𝐵) ≠ 1))
64	63	baib 535	. . . . . . . . 9 ⊢ ((𝐴 gcd 𝐵) ∈ ℕ → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ≠ 1))
65	62, 64	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (𝐴 gcd 𝐵) ≠ 1))
66	60, 65	sylibrd 259	. . . . . . 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 34	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) → (𝐴 gcd 𝐵) ∈ (ℤ_≥‘2)))
68		exprmfct 16728	. . . . . . 7 ⊢ ((𝐴 gcd 𝐵) ∈ (ℤ_≥‘2) → ∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵))
69	62	nnzd 12620	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
70	69	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∈ ℤ)
71		gcddvds 16527	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
72	41, 21, 71	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
73	72	simpld 494	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐴)
74		iddvdsexp 16304	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∥ (𝐴↑𝑁))
75	41, 42, 74	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∥ (𝐴↑𝑁))
76	70, 41, 19, 73, 75	dvdstrd 16319	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ (𝐴↑𝑁))
77		dvdstr 16318	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 694	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ (𝐴↑𝑁)))
80	72	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) ∥ 𝐵)
81		dvdstr 16318	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 694	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → 𝑝 ∥ 𝐵))
84	79, 83	jcad 512	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝐴 gcd 𝐵) → (𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵)))
85		dvdsgcd 16568	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑝 ∥ (𝐴↑𝑁) ∧ 𝑝 ∥ 𝐵) → 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
86	26, 19, 21, 85	syl3anc 1373	. . . . . . . . . 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 5128	. . . . . . . . . . . . . 14 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ 𝑝 ∥ 1))
88	87	notbid 318	. . . . . . . . . . . . 13 ⊢ (((𝐴↑𝑁) gcd 𝐵) = 1 → (¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) ↔ ¬ 𝑝 ∥ 1))
89	53, 88	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → (((𝐴↑𝑁) gcd 𝐵) = 1 → ¬ 𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵)))
90	89	necon2ad 2948	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → (𝑝 ∥ ((𝐴↑𝑁) gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
91	90	adantl 481	. . . . . . . . . 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 60	. . . . . . . . 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 3142	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (∃𝑝 ∈ ℙ 𝑝 ∥ (𝐴 gcd 𝐵) → ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
94		eluz2b3 12943	. . . . . . . . . 10 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ ∧ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
95	94	baib 535	. . . . . . . . 9 ⊢ (((𝐴↑𝑁) gcd 𝐵) ∈ ℕ → (((𝐴↑𝑁) gcd 𝐵) ∈ (ℤ_≥‘2) ↔ ((𝐴↑𝑁) gcd 𝐵) ≠ 1))
96	34, 95	syl 17	. . . . . . . 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 259	. . . . . . 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 34	. . . . . 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 212	. . . . 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 309	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) ≠ 1 ↔ (𝐴 gcd 𝐵) ≠ 1))
101	100	necon4bid 2978	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (((𝐴↑𝑁) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1))
102	101	ex 412	. 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 179	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∃wrex 3061 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 0cc0 11134 1c1 11135 ℕcn 12245 2c2 12300 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ↑cexp 14084 ∥ cdvds 16277 gcd cgcd 16518 ℙcprime 16695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-fz 13530 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-gcd 16519 df-prm 16696

This theorem is referenced by: rpexp1i 16747 phiprmpw 16800 pockthlem 16930 logbgcd1irr 26761 aks4d1p8d3 42104 hashscontpow1 42139 flt4lem7 42657

W3C validator