Theorem prmirredlem 21389

Description: A positive integer is irreducible over ℤ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)

Hypothesis

Ref	Expression
prmirred.i	⊢ 𝐼 = (Irred‘ℤring)

Assertion

Ref	Expression
prmirredlem	⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))

Proof of Theorem prmirredlem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zringring 21366	. . . . . 6 ⊢ ℤring ∈ Ring
2		prmirred.i	. . . . . . 7 ⊢ 𝐼 = (Irred‘ℤring)
3		zring1 21376	. . . . . . 7 ⊢ 1 = (1r‘ℤring)
4	2, 3	irredn1 20342	. . . . . 6 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 1)
5	1, 4	mpan 690	. . . . 5 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 1)
6	5	anim2i 617	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7		eluz2b3 12888	. . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
8	6, 7	sylibr 234	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ (ℤ≥‘2))
9		nnz 12557	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
10	9	ad2antrl 728	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℤ)
11		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∥ 𝐴)
12		nnne0 12227	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
13	12	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ≠ 0)
14		nnz 12557	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
15	14	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℤ)
16		dvdsval2 16232	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦 ∥ 𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
17	10, 13, 15, 16	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∥ 𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
18	11, 17	mpbid 232	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝐴 / 𝑦) ∈ ℤ)
19	15	zcnd 12646	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℂ)
20		nncn 12201	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
21	20	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℂ)
22	19, 21, 13	divcan2d 11967	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23		simplr 768	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ 𝐼)
24	22, 23	eqeltrd 2829	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25		zringbas 21370	. . . . . . . 8 ⊢ ℤ = (Base‘ℤring)
26		eqid 2730	. . . . . . . 8 ⊢ (Unit‘ℤring) = (Unit‘ℤring)
27		zringmulr 21374	. . . . . . . 8 ⊢ · = (.r‘ℤring)
28	2, 25, 26, 27	irredmul 20345	. . . . . . 7 ⊢ ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
29	10, 18, 24, 28	syl3anc 1373	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30		zringunit 21383	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Unit‘ℤring) ↔ (𝑦 ∈ ℤ ∧ (abs‘𝑦) = 1))
31	30	baib 535	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
32	10, 31	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
33		nnnn0 12456	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34		nn0re 12458	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
35		nn0ge0 12474	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
36	34, 35	absidd 15396	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
37	33, 36	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
38	37	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘𝑦) = 𝑦)
39	38	eqeq1d 2732	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘𝑦) = 1 ↔ 𝑦 = 1))
40	32, 39	bitrd 279	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ 𝑦 = 1))
41		zringunit 21383	. . . . . . . . . 10 ⊢ ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ ((𝐴 / 𝑦) ∈ ℤ ∧ (abs‘(𝐴 / 𝑦)) = 1))
42	41	baib 535	. . . . . . . . 9 ⊢ ((𝐴 / 𝑦) ∈ ℤ → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ (abs‘(𝐴 / 𝑦)) = 1))
43	18, 42	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ (abs‘(𝐴 / 𝑦)) = 1))
44		nnre 12200	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
45	44	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℝ)
46		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℕ)
47	45, 46	nndivred 12247	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48		nnnn0 12456	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49		nn0ge0 12474	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
51	50	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 ≤ 𝐴)
52	46	nnred 12208	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℝ)
53		nngt0 12224	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
54	53	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 < 𝑦)
55		divge0 12059	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
56	45, 51, 52, 54, 55	syl22anc 838	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 ≤ (𝐴 / 𝑦))
57	47, 56	absidd 15396	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
58	57	eqeq1d 2732	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59		1cnd 11176	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 1 ∈ ℂ)
60	19, 21, 59, 13	divmuld 11987	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
61	21	mulridd 11198	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · 1) = 𝑦)
62	61	eqeq1d 2732	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝑦 · 1) = 𝐴 ↔ 𝑦 = 𝐴))
63	58, 60, 62	3bitrd 305	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ 𝑦 = 𝐴))
64	43, 63	bitrd 279	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ 𝑦 = 𝐴))
65	40, 64	orbi12d 918	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)) ↔ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
66	29, 65	mpbid 232	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 = 1 ∨ 𝑦 = 𝐴))
67	66	expr 456	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ 𝑦 ∈ ℕ) → (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
68	67	ralrimiva 3126	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69		isprm2 16659	. . 3 ⊢ (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ≥‘2) ∧ ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
70	8, 68, 69	sylanbrc 583	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ℙ)
71		prmz 16652	. . . 4 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72		1nprm 16656	. . . . 5 ⊢ ¬ 1 ∈ ℙ
73		zringunit 21383	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74		prmnn 16651	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75		nn0re 12458	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
76	75, 49	absidd 15396	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0 → (abs‘𝐴) = 𝐴)
77	74, 48, 76	3syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ ℙ → (abs‘𝐴) = 𝐴)
78		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℙ)
79	77, 78	eqeltrd 2829	. . . . . . . 8 ⊢ (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80		eleq1 2817	. . . . . . . 8 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) ∈ ℙ ↔ 1 ∈ ℙ))
81	79, 80	syl5ibcom 245	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → ((abs‘𝐴) = 1 → 1 ∈ ℙ))
82	81	adantld 490	. . . . . 6 ⊢ (𝐴 ∈ ℙ → ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℙ))
83	73, 82	biimtrid 242	. . . . 5 ⊢ (𝐴 ∈ ℙ → (𝐴 ∈ (Unit‘ℤring) → 1 ∈ ℙ))
84	72, 83	mtoi 199	. . . 4 ⊢ (𝐴 ∈ ℙ → ¬ 𝐴 ∈ (Unit‘ℤring))
85		dvdsmul1 16254	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
86	85	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ (𝑥 · 𝑦))
87		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) = 𝐴)
88	86, 87	breqtrd 5136	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ 𝐴)
89		simplrl 776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
90	71	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
91		absdvdsb 16251	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
92	89, 90, 91	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
93	88, 92	mpbid 232	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∥ 𝐴)
94		breq1 5113	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95		eqeq1 2734	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96		eqeq1 2734	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
97	95, 96	orbi12d 918	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑥) → ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
98	94, 97	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = (abs‘𝑥) → ((𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
99	69	simprbi 496	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
100	99	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
101	89	zcnd 12646	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
102	74	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
103	102	nnne0d 12243	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ≠ 0)
104		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℤ)
105	104	zcnd 12646	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106	105	mul02d 11379	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107	103, 87, 106	3netr4d 3003	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108		oveq1 7397	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109	108	necon3i 2958	. . . . . . . . . . . . 13 ⊢ ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
110	107, 109	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
111	101, 110	absne0d 15423	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112	111	neneqd 2931	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113		nn0abscl 15285	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
114	89, 113	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
115		elnn0 12451	. . . . . . . . . . . 12 ⊢ ((abs‘𝑥) ∈ ℕ0 ↔ ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
116	114, 115	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
117	116	ord 864	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (¬ (abs‘𝑥) ∈ ℕ → (abs‘𝑥) = 0))
118	112, 117	mt3d 148	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ)
119	98, 100, 118	rspcdva 3592	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
120	93, 119	mpd 15	. . . . . . 7 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))
121		zringunit 21383	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Unit‘ℤring) ↔ (𝑥 ∈ ℤ ∧ (abs‘𝑥) = 1))
122	121	baib 535	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
123	89, 122	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
124	104, 31	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
125	105	abscld 15412	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126	125	recnd 11209	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127		1cnd 11176	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128	101	abscld 15412	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129	128	recnd 11209	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130	126, 127, 129, 111	mulcand 11818	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
131	87	fveq2d 6865	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132	101, 105	absmuld 15430	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
133	77	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134	131, 132, 133	3eqtr3d 2773	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135	129	mulridd 11198	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136	134, 135	eqeq12d 2746	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137		eqcom 2737	. . . . . . . . . 10 ⊢ (𝐴 = (abs‘𝑥) ↔ (abs‘𝑥) = 𝐴)
138	136, 137	bitrdi 287	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑥) = 𝐴))
139	124, 130, 138	3bitr2d 307	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 𝐴))
140	123, 139	orbi12d 918	. . . . . . 7 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
141	120, 140	mpbird 257	. . . . . 6 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))
142	141	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
143	142	ralrimivva 3181	. . . 4 ⊢ (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
144	25, 26, 2, 27	isirred2 20337	. . . 4 ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ (Unit‘ℤring) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))))
145	71, 84, 143, 144	syl3anbrc 1344	. . 3 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ 𝐼)
146	145	adantl 481	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴 ∈ 𝐼)
147	70, 146	impbida 800	1 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   · cmul 11080   < clt 11215   ≤ cle 11216   / cdiv 11842  ℕcn 12193  2c2 12248  ℕ₀cn0 12449  ℤcz 12536  ℤ_≥cuz 12800  abscabs 15207   ∥ cdvds 16229  ℙcprime 16648  Ringcrg 20149  Unitcui 20271  Irredcir 20272  ℤ_ringczring 21363

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154  ax-mulf 11155

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-rp 12959  df-fz 13476  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16230  df-prm 16649  df-gz 16908  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-subg 19062  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-irred 20275  df-invr 20304  df-dvr 20317  df-subrng 20462  df-subrg 20486  df-drng 20647  df-cnfld 21272  df-zring 21364

This theorem is referenced by:  dfprm2  21390  prmirred  21391