Theorem prmirredlem 21466

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 21443	. . . . . 6 ⊢ ℤring ∈ Ring
2		prmirred.i	. . . . . . 7 ⊢ 𝐼 = (Irred‘ℤring)
3		zring1 21453	. . . . . . 7 ⊢ 1 = (1r‘ℤring)
4	2, 3	irredn1 20401	. . . . . 6 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 1)
5	1, 4	mpan 691	. . . . 5 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 1)
6	5	anim2i 618	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7		eluz2b3 12867	. . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
8	6, 7	sylibr 234	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ (ℤ≥‘2))
9		nnz 12540	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
10	9	ad2antrl 729	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℤ)
11		simprr 773	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∥ 𝐴)
12		nnne0 12206	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
13	12	ad2antrl 729	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ≠ 0)
14		nnz 12540	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
15	14	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℤ)
16		dvdsval2 16219	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦 ∥ 𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
17	10, 13, 15, 16	syl3anc 1374	. . . . . . . 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 12629	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℂ)
20		nncn 12177	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
21	20	ad2antrl 729	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℂ)
22	19, 21, 13	divcan2d 11928	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23		simplr 769	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ 𝐼)
24	22, 23	eqeltrd 2837	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25		zringbas 21447	. . . . . . . 8 ⊢ ℤ = (Base‘ℤring)
26		eqid 2737	. . . . . . . 8 ⊢ (Unit‘ℤring) = (Unit‘ℤring)
27		zringmulr 21451	. . . . . . . 8 ⊢ · = (.r‘ℤring)
28	2, 25, 26, 27	irredmul 20404	. . . . . . 7 ⊢ ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
29	10, 18, 24, 28	syl3anc 1374	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30		zringunit 21460	. . . . . . . . . 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 12439	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34		nn0re 12441	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
35		nn0ge0 12457	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
36	34, 35	absidd 15380	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
37	33, 36	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
38	37	ad2antrl 729	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘𝑦) = 𝑦)
39	38	eqeq1d 2739	. . . . . . . 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 21460	. . . . . . . . . 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 12176	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
45	44	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℝ)
46		simprl 771	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℕ)
47	45, 46	nndivred 12226	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48		nnnn0 12439	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49		nn0ge0 12457	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
51	50	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 ≤ 𝐴)
52	46	nnred 12184	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℝ)
53		nngt0 12203	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
54	53	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 < 𝑦)
55		divge0 12020	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
56	45, 51, 52, 54, 55	syl22anc 839	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 ≤ (𝐴 / 𝑦))
57	47, 56	absidd 15380	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
58	57	eqeq1d 2739	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59		1cnd 11134	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 1 ∈ ℂ)
60	19, 21, 59, 13	divmuld 11948	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
61	21	mulridd 11157	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · 1) = 𝑦)
62	61	eqeq1d 2739	. . . . . . . . 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 919	. . . . . 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 3130	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69		isprm2 16646	. . 3 ⊢ (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ≥‘2) ∧ ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
70	8, 68, 69	sylanbrc 584	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ℙ)
71		prmz 16639	. . . 4 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72		1nprm 16643	. . . . 5 ⊢ ¬ 1 ∈ ℙ
73		zringunit 21460	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74		prmnn 16638	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75		nn0re 12441	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
76	75, 49	absidd 15380	. . . . . . . . . 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 2837	. . . . . . . 8 ⊢ (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80		eleq1 2825	. . . . . . . 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 16241	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
86	85	ad2antlr 728	. . . . . . . . . 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 5112	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ 𝐴)
89		simplrl 777	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
90	71	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
91		absdvdsb 16238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
92	89, 90, 91	syl2anc 585	. . . . . . . . 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 5089	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95		eqeq1 2741	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96		eqeq1 2741	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
97	95, 96	orbi12d 919	. . . . . . . . . 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 497	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
100	99	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
101	89	zcnd 12629	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
102	74	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
103	102	nnne0d 12222	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ≠ 0)
104		simplrr 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℤ)
105	104	zcnd 12629	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106	105	mul02d 11339	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107	103, 87, 106	3netr4d 3010	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108		oveq1 7369	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109	108	necon3i 2965	. . . . . . . . . . . . 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 15407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112	111	neneqd 2938	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113		nn0abscl 15269	. . . . . . . . . . . . 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 12434	. . . . . . . . . . . 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 865	. . . . . . . . . 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 3566	. . . . . . . 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 21460	. . . . . . . . . 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 15396	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126	125	recnd 11168	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127		1cnd 11134	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128	101	abscld 15396	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129	128	recnd 11168	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130	126, 127, 129, 111	mulcand 11778	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
131	87	fveq2d 6840	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132	101, 105	absmuld 15414	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
133	77	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134	131, 132, 133	3eqtr3d 2780	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135	129	mulridd 11157	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136	134, 135	eqeq12d 2753	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137		eqcom 2744	. . . . . . . . . 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 919	. . . . . . 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 20396	. . . 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 1345	. . 3 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ 𝐼)
146	145	adantl 481	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴 ∈ 𝐼)
147	70, 146	impbida 801	1 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362  ℂcc 11031  ℝcr 11032  0cc0 11033  1c1 11034   · cmul 11038   < clt 11174   ≤ cle 11175   / cdiv 11802  ℕcn 12169  2c2 12231  ℕ₀cn0 12432  ℤcz 12519  ℤ_≥cuz 12783  abscabs 15191   ∥ cdvds 16216  ℙcprime 16635  Ringcrg 20209  Unitcui 20330  Irredcir 20331  ℤ_ringczring 21440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111  ax-addf 11112  ax-mulf 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-tpos 8171  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-rp 12938  df-fz 13457  df-seq 13959  df-exp 14019  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-dvds 16217  df-prm 16636  df-gz 16896  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-starv 17230  df-tset 17234  df-ple 17235  df-ds 17237  df-unif 17238  df-0g 17399  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-subg 19094  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-cring 20212  df-oppr 20312  df-dvdsr 20332  df-unit 20333  df-irred 20334  df-invr 20363  df-dvr 20376  df-subrng 20518  df-subrg 20542  df-drng 20703  df-cnfld 21349  df-zring 21441

This theorem is referenced by:  dfprm2  21467  prmirred  21468