Theorem prmirredlem 21778

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 21755	. . . . . 6 ⊢ ℤring ∈ Ring
2		prmirred.i	. . . . . . 7 ⊢ 𝐼 = (Irred‘ℤring)
3		zring1 21765	. . . . . . 7 ⊢ 1 = (1r‘ℤring)
4	2, 3	irredn1 20656	. . . . . 6 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 1)
5	1, 4	mpan 703	. . . . 5 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 1)
6	5	anim2i 629	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7		eluz2b3 13049	. . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
8	6, 7	sylibr 237	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ (ℤ≥‘2))
9		nnz 12714	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
10	9	ad2antrl 741	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℤ)
11		simprr 785	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∥ 𝐴)
12		nnne0 12372	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
13	12	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ≠ 0)
14		nnz 12714	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
15	14	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℤ)
16		dvdsval2 16425	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦 ∥ 𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
17	10, 13, 15, 16	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∥ 𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
18	11, 17	mpbid 235	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝐴 / 𝑦) ∈ ℤ)
19	15	zcnd 12804	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℂ)
20		nncn 12343	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
21	20	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℂ)
22	19, 21, 13	divcan2d 12095	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ 𝐼)
24	22, 23	eqeltrd 2861	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25		zringbas 21759	. . . . . . . 8 ⊢ ℤ = (Base‘ℤring)
26		eqid 2761	. . . . . . . 8 ⊢ (Unit‘ℤring) = (Unit‘ℤring)
27		zringmulr 21763	. . . . . . . 8 ⊢ · = (.r‘ℤring)
28	2, 25, 26, 27	irredmul 20659	. . . . . . 7 ⊢ ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
29	10, 18, 24, 28	syl3anc 1398	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30		zringunit 21772	. . . . . . . . . 10 ⊢ (𝑦 ∈ (Unit‘ℤring) ↔ (𝑦 ∈ ℤ ∧ (abs‘𝑦) = 1))
31	30	baib 545	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
32	10, 31	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
33		nnnn0 12613	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34		nn0re 12615	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
35		nn0ge0 12631	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
36	34, 35	absidd 15590	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
37	33, 36	syl 18	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
38	37	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘𝑦) = 𝑦)
39	38	eqeq1d 2763	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘𝑦) = 1 ↔ 𝑦 = 1))
40	32, 39	bitrd 282	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ 𝑦 = 1))
41		zringunit 21772	. . . . . . . . . 10 ⊢ ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ ((𝐴 / 𝑦) ∈ ℤ ∧ (abs‘(𝐴 / 𝑦)) = 1))
42	41	baib 545	. . . . . . . . 9 ⊢ ((𝐴 / 𝑦) ∈ ℤ → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ (abs‘(𝐴 / 𝑦)) = 1))
43	18, 42	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ (abs‘(𝐴 / 𝑦)) = 1))
44		nnre 12342	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
45	44	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝐴 ∈ ℝ)
46		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℕ)
47	45, 46	nndivred 12392	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48		nnnn0 12613	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49		nn0ge0 12631	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
50	48, 49	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
51	50	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 ≤ 𝐴)
52	46	nnred 12350	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 𝑦 ∈ ℝ)
53		nngt0 12369	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
54	53	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 < 𝑦)
55		divge0 12186	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
56	45, 51, 52, 54, 55	syl22anc 852	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 0 ≤ (𝐴 / 𝑦))
57	47, 56	absidd 15590	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
58	57	eqeq1d 2763	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59		1cnd 11302	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → 1 ∈ ℂ)
60	19, 21, 59, 13	divmuld 12115	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
61	21	mulridd 11326	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 · 1) = 𝑦)
62	61	eqeq1d 2763	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝑦 · 1) = 𝐴 ↔ 𝑦 = 𝐴))
63	58, 60, 62	3bitrd 308	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ 𝑦 = 𝐴))
64	43, 63	bitrd 282	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ 𝑦 = 𝐴))
65	40, 64	orbi12d 932	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → ((𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)) ↔ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
66	29, 65	mpbid 235	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦 ∥ 𝐴)) → (𝑦 = 1 ∨ 𝑦 = 𝐴))
67	66	expr 462	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) ∧ 𝑦 ∈ ℕ) → (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
68	67	ralrimiva 3155	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69		isprm2 16857	. . 3 ⊢ (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ≥‘2) ∧ ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
70	8, 68, 69	sylanbrc 595	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ ℙ)
71		prmz 16850	. . . 4 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72		1nprm 16854	. . . . 5 ⊢ ¬ 1 ∈ ℙ
73		zringunit 21772	. . . . . 6 ⊢ (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74		prmnn 16849	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75		nn0re 12615	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ)
76	75, 49	absidd 15590	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0 → (abs‘𝐴) = 𝐴)
77	74, 48, 76	3syl 19	. . . . . . . . 9 ⊢ (𝐴 ∈ ℙ → (abs‘𝐴) = 𝐴)
78		id 23	. . . . . . . . 9 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℙ)
79	77, 78	eqeltrd 2861	. . . . . . . 8 ⊢ (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80		eleq1 2849	. . . . . . . 8 ⊢ ((abs‘𝐴) = 1 → ((abs‘𝐴) ∈ ℙ ↔ 1 ∈ ℙ))
81	79, 80	syl5ibcom 248	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → ((abs‘𝐴) = 1 → 1 ∈ ℙ))
82	81	adantld 496	. . . . . 6 ⊢ (𝐴 ∈ ℙ → ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℙ))
83	73, 82	biimtrid 245	. . . . 5 ⊢ (𝐴 ∈ ℙ → (𝐴 ∈ (Unit‘ℤring) → 1 ∈ ℙ))
84	72, 83	mtoi 202	. . . 4 ⊢ (𝐴 ∈ ℙ → ¬ 𝐴 ∈ (Unit‘ℤring))
85		dvdsmul1 16447	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
86	85	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ (𝑥 · 𝑦))
87		simpr 490	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) = 𝐴)
88	86, 87	breqtrd 5131	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ 𝐴)
89		simplrl 789	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
90	71	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
91		absdvdsb 16444	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
92	89, 90, 91	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
93	88, 92	mpbid 235	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∥ 𝐴)
94		breq1 5106	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 ∥ 𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95		eqeq1 2765	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96		eqeq1 2765	. . . . . . . . . . 11 ⊢ (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
97	95, 96	orbi12d 932	. . . . . . . . . 10 ⊢ (𝑦 = (abs‘𝑥) → ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
98	94, 97	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = (abs‘𝑥) → ((𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
99	69	simprbi 503	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
100	99	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦 ∥ 𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
101	89	zcnd 12804	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
102	74	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
103	102	nnne0d 12388	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ≠ 0)
104		simplrr 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℤ)
105	104	zcnd 12804	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106	105	mul02d 11508	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107	103, 87, 106	3netr4d 3033	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109	108	necon3i 2988	. . . . . . . . . . . . 13 ⊢ ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
110	107, 109	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
111	101, 110	absne0d 15617	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112	111	neneqd 2961	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113		nn0abscl 15479	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
114	89, 113	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
115		elnn0 12608	. . . . . . . . . . . 12 ⊢ ((abs‘𝑥) ∈ ℕ0 ↔ ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
116	114, 115	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
117	116	ord 878	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (¬ (abs‘𝑥) ∈ ℕ → (abs‘𝑥) = 0))
118	112, 117	mt3d 149	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ)
119	98, 100, 118	rspcdva 3578	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
120	93, 119	mpd 16	. . . . . . 7 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))
121		zringunit 21772	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Unit‘ℤring) ↔ (𝑥 ∈ ℤ ∧ (abs‘𝑥) = 1))
122	121	baib 545	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
123	89, 122	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
124	104, 31	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
125	105	abscld 15606	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126	125	recnd 11337	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127		1cnd 11302	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128	101	abscld 15606	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129	128	recnd 11337	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130	126, 127, 129, 111	mulcand 11949	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
131	87	fveq2d 6889	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132	101, 105	absmuld 15624	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
133	77	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134	131, 132, 133	3eqtr3d 2804	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135	129	mulridd 11326	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136	134, 135	eqeq12d 2777	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137		eqcom 2768	. . . . . . . . . 10 ⊢ (𝐴 = (abs‘𝑥) ↔ (abs‘𝑥) = 𝐴)
138	136, 137	bitrdi 290	. . . . . . . . 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 310	. . . . . . . 8 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 𝐴))
140	123, 139	orbi12d 932	. . . . . . 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 260	. . . . . 6 ⊢ (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))
142	141	ex 418	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
143	142	ralrimivva 3206	. . . 4 ⊢ (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
144	25, 26, 2, 27	isirred2 20651	. . . 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 1362	. . 3 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ 𝐼)
146	145	adantl 487	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴 ∈ 𝐼)
147	70, 146	impbida 813	1 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  ℂcc 11198  ℝcr 11199  0cc0 11200  1c1 11201   · cmul 11205   < clt 11343   ≤ cle 11344   / cdiv 11973  ℕcn 12335  2c2 12397  ℕ₀cn0 12606  ℤcz 12693  ℤ_≥cuz 12965  abscabs 15401   ∥ cdvds 16422  ℙcprime 16846  Ringcrg 20459  Unitcui 20585  Irredcir 20586  ℤ_ringczring 21752

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279  ax-mulf 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-rp 13121  df-fz 13640  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-dvds 16423  df-prm 16847  df-gz 17108  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-subg 19333  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-irred 20589  df-invr 20618  df-dvr 20631  df-subrng 20798  df-subrg 20822  df-drng 20982  df-cnfld 21679  df-zring 21753

This theorem is used by:  dfprm2  21779  prmirred  21780