Theorem prmirredlem 21414

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 21391	. . . . . 6 ⊢ ℤring ∈ Ring
2		prmirred.i	. . . . . . 7 ⊢ 𝐼 = (Irred‘ℤring)
3		zring1 21401	. . . . . . 7 ⊢ 1 = (1r‘ℤring)
4	2, 3	irredn1 20346	. . . . . 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 12857	. . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
8	6, 7	sylibr 234	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ (ℤ≥‘2))
9		nnz 12526	. . . . . . . 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 12196	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
13	12	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ≠ 0)
14		nnz 12526	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
15	14	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℤ)
16		dvdsval2 16201	. . . . . . . . 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 12615	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℂ)
20		nncn 12170	. . . . . . . . . 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 11936	. . . . . . . 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 2828	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25		zringbas 21395	. . . . . . . 8 ⊢ ℤ = (Base‘ℤring)
26		eqid 2729	. . . . . . . 8 ⊢ (Unit‘ℤring) = (Unit‘ℤring)
27		zringmulr 21399	. . . . . . . 8 ⊢ · = (.r‘ℤring)
28	2, 25, 26, 27	irredmul 20349	. . . . . . 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 21408	. . . . . . . . . 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 12425	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34		nn0re 12427	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
35		nn0ge0 12443	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
36	34, 35	absidd 15365	. . . . . . . . . . 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 2731	. . . . . . . 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 21408	. . . . . . . . . 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 12169	. . . . . . . . . . . . 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 12216	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48		nnnn0 12425	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49		nn0ge0 12443	. . . . . . . . . . . . . 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 12177	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℝ)
53		nngt0 12193	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
54	53	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 < 𝑦)
55		divge0 12028	. . . . . . . . . . . 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 15365	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
58	57	eqeq1d 2731	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59		1cnd 11145	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 1 ∈ ℂ)
60	19, 21, 59, 13	divmuld 11956	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
61	21	mulridd 11167	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · 1) = 𝑦)
62	61	eqeq1d 2731	. . . . . . . . 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 3125	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69		isprm2 16628	. . 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 16621	. . . 4 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72		1nprm 16625	. . . . 5 ⊢ ¬ 1 ∈ ℙ
73		zringunit 21408	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74		prmnn 16620	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75		nn0re 12427	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
76	75, 49	absidd 15365	. . . . . . . . . 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 2828	. . . . . . . 8 ⊢ (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80		eleq1 2816	. . . . . . . 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 16223	. . . . . . . . . . 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 5128	. . . . . . . . 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 16220	. . . . . . . . . 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 5105	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96		eqeq1 2733	. . . . . . . . . . 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 12615	. . . . . . . . . . . 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 12212	. . . . . . . . . . . . . 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 12615	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106	105	mul02d 11348	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107	103, 87, 106	3netr4d 3002	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108		oveq1 7376	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109	108	necon3i 2957	. . . . . . . . . . . . 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 15392	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112	111	neneqd 2930	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113		nn0abscl 15254	. . . . . . . . . . . . 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 12420	. . . . . . . . . . . 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 3586	. . . . . . . 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 21408	. . . . . . . . . 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 15381	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126	125	recnd 11178	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127		1cnd 11145	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128	101	abscld 15381	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129	128	recnd 11178	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130	126, 127, 129, 111	mulcand 11787	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
131	87	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132	101, 105	absmuld 15399	. . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135	129	mulridd 11167	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136	134, 135	eqeq12d 2745	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137		eqcom 2736	. . . . . . . . . 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 3178	. . . 4 ⊢ (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
144	25, 26, 2, 27	isirred2 20341	. . . 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 2925  ∀wral 3044   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   · cmul 11049   < clt 11184   ≤ cle 11185   / cdiv 11811  ℕcn 12162  2c2 12217  ℕ₀cn0 12418  ℤcz 12505  ℤ_≥cuz 12769  abscabs 15176   ∥ cdvds 16198  ℙcprime 16617  Ringcrg 20153  Unitcui 20275  Irredcir 20276  ℤ_ringczring 21388

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123  ax-mulf 11124

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-rp 12928  df-fz 13445  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-dvds 16199  df-prm 16618  df-gz 16877  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-0g 17380  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-grp 18850  df-minusg 18851  df-subg 19037  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-cring 20156  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-irred 20279  df-invr 20308  df-dvr 20321  df-subrng 20466  df-subrg 20490  df-drng 20651  df-cnfld 21297  df-zring 21389

This theorem is referenced by:  dfprm2  21415  prmirred  21416