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Theorem prmirredlem 21483
Description: A positive integer is irreducible over 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 → (𝐴𝐼𝐴 ∈ ℙ))

Proof of Theorem prmirredlem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zringring 21460 . . . . . 6 ring ∈ Ring
2 prmirred.i . . . . . . 7 𝐼 = (Irred‘ℤring)
3 zring1 21470 . . . . . . 7 1 = (1r‘ℤring)
42, 3irredn1 20426 . . . . . 6 ((ℤring ∈ Ring ∧ 𝐴𝐼) → 𝐴 ≠ 1)
51, 4mpan 690 . . . . 5 (𝐴𝐼𝐴 ≠ 1)
65anim2i 617 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7 eluz2b3 12964 . . . 4 (𝐴 ∈ (ℤ‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
86, 7sylibr 234 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ (ℤ‘2))
9 nnz 12634 . . . . . . . 8 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
109ad2antrl 728 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℤ)
11 simprr 773 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦𝐴)
12 nnne0 12300 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1312ad2antrl 728 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ≠ 0)
14 nnz 12634 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
1514ad2antrr 726 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℤ)
16 dvdsval2 16293 . . . . . . . . 9 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1710, 13, 15, 16syl3anc 1373 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1811, 17mpbid 232 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℤ)
1915zcnd 12723 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℂ)
20 nncn 12274 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2120ad2antrl 728 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℂ)
2219, 21, 13divcan2d 12045 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23 simplr 769 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴𝐼)
2422, 23eqeltrd 2841 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25 zringbas 21464 . . . . . . . 8 ℤ = (Base‘ℤring)
26 eqid 2737 . . . . . . . 8 (Unit‘ℤring) = (Unit‘ℤring)
27 zringmulr 21468 . . . . . . . 8 · = (.r‘ℤring)
282, 25, 26, 27irredmul 20429 . . . . . . 7 ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
2910, 18, 24, 28syl3anc 1373 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30 zringunit 21477 . . . . . . . . . 10 (𝑦 ∈ (Unit‘ℤring) ↔ (𝑦 ∈ ℤ ∧ (abs‘𝑦) = 1))
3130baib 535 . . . . . . . . 9 (𝑦 ∈ ℤ → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
3210, 31syl 17 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
33 nnnn0 12533 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34 nn0re 12535 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
35 nn0ge0 12551 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
3634, 35absidd 15461 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
3733, 36syl 17 . . . . . . . . . 10 (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
3837ad2antrl 728 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘𝑦) = 𝑦)
3938eqeq1d 2739 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘𝑦) = 1 ↔ 𝑦 = 1))
4032, 39bitrd 279 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ 𝑦 = 1))
41 zringunit 21477 . . . . . . . . . 10 ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ ((𝐴 / 𝑦) ∈ ℤ ∧ (abs‘(𝐴 / 𝑦)) = 1))
4241baib 535 . . . . . . . . 9 ((𝐴 / 𝑦) ∈ ℤ → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ (abs‘(𝐴 / 𝑦)) = 1))
4318, 42syl 17 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ (abs‘(𝐴 / 𝑦)) = 1))
44 nnre 12273 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
4544ad2antrr 726 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℝ)
46 simprl 771 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℕ)
4745, 46nndivred 12320 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48 nnnn0 12533 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49 nn0ge0 12551 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
5048, 49syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
5150ad2antrr 726 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ 𝐴)
5246nnred 12281 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℝ)
53 nngt0 12297 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ → 0 < 𝑦)
5453ad2antrl 728 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 < 𝑦)
55 divge0 12137 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
5645, 51, 52, 54, 55syl22anc 839 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ (𝐴 / 𝑦))
5747, 56absidd 15461 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
5857eqeq1d 2739 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59 1cnd 11256 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 1 ∈ ℂ)
6019, 21, 59, 13divmuld 12065 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
6121mulridd 11278 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · 1) = 𝑦)
6261eqeq1d 2739 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝑦 · 1) = 𝐴𝑦 = 𝐴))
6358, 60, 623bitrd 305 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ 𝑦 = 𝐴))
6443, 63bitrd 279 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ 𝑦 = 𝐴))
6540, 64orbi12d 919 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)) ↔ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6629, 65mpbid 232 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 = 1 ∨ 𝑦 = 𝐴))
6766expr 456 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ 𝑦 ∈ ℕ) → (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6867ralrimiva 3146 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69 isprm2 16719 . . 3 (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ‘2) ∧ ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
708, 68, 69sylanbrc 583 . 2 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ ℙ)
71 prmz 16712 . . . 4 (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72 1nprm 16716 . . . . 5 ¬ 1 ∈ ℙ
73 zringunit 21477 . . . . . 6 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74 prmnn 16711 . . . . . . . . . 10 (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75 nn0re 12535 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
7675, 49absidd 15461 . . . . . . . . . 10 (𝐴 ∈ ℕ0 → (abs‘𝐴) = 𝐴)
7774, 48, 763syl 18 . . . . . . . . 9 (𝐴 ∈ ℙ → (abs‘𝐴) = 𝐴)
78 id 22 . . . . . . . . 9 (𝐴 ∈ ℙ → 𝐴 ∈ ℙ)
7977, 78eqeltrd 2841 . . . . . . . 8 (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80 eleq1 2829 . . . . . . . 8 ((abs‘𝐴) = 1 → ((abs‘𝐴) ∈ ℙ ↔ 1 ∈ ℙ))
8179, 80syl5ibcom 245 . . . . . . 7 (𝐴 ∈ ℙ → ((abs‘𝐴) = 1 → 1 ∈ ℙ))
8281adantld 490 . . . . . 6 (𝐴 ∈ ℙ → ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℙ))
8373, 82biimtrid 242 . . . . 5 (𝐴 ∈ ℙ → (𝐴 ∈ (Unit‘ℤring) → 1 ∈ ℙ))
8472, 83mtoi 199 . . . 4 (𝐴 ∈ ℙ → ¬ 𝐴 ∈ (Unit‘ℤring))
85 dvdsmul1 16315 . . . . . . . . . . 11 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
8685ad2antlr 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)
8886, 87breqtrd 5169 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥𝐴)
89 simplrl 777 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
9071ad2antrr 726 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
91 absdvdsb 16312 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
9289, 90, 91syl2anc 584 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
9388, 92mpbid 232 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∥ 𝐴)
94 breq1 5146 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → (𝑦𝐴 ↔ (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))
9795, 96orbi12d 919 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
9894, 97imbi12d 344 . . . . . . . . 9 (𝑦 = (abs‘𝑥) → ((𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
9969simprbi 496 . . . . . . . . . 10 (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10099ad2antrr 726 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10189zcnd 12723 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
10274ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
103102nnne0d 12316 . . . . . . . . . . . . . 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)
105104zcnd 12723 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106105mul02d 11459 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107103, 87, 1063netr4d 3018 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108 oveq1 7438 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109108necon3i 2973 . . . . . . . . . . . . 13 ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
111101, 110absne0d 15486 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112111neneqd 2945 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113 nn0abscl 15351 . . . . . . . . . . . . 13 (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
11489, 113syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
115 elnn0 12528 . . . . . . . . . . . 12 ((abs‘𝑥) ∈ ℕ0 ↔ ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
116114, 115sylib 218 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
117116ord 865 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (¬ (abs‘𝑥) ∈ ℕ → (abs‘𝑥) = 0))
118112, 117mt3d 148 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ)
11998, 100, 118rspcdva 3623 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
12093, 119mpd 15 . . . . . . 7 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))
121 zringunit 21477 . . . . . . . . . 10 (𝑥 ∈ (Unit‘ℤring) ↔ (𝑥 ∈ ℤ ∧ (abs‘𝑥) = 1))
122121baib 535 . . . . . . . . 9 (𝑥 ∈ ℤ → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
12389, 122syl 17 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
124104, 31syl 17 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
125105abscld 15475 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126125recnd 11289 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127 1cnd 11256 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128101abscld 15475 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129128recnd 11289 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130126, 127, 129, 111mulcand 11896 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
13187fveq2d 6910 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132101, 105absmuld 15493 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
13377ad2antrr 726 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134131, 132, 1333eqtr3d 2785 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135129mulridd 11278 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136134, 135eqeq12d 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)
138136, 137bitrdi 287 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑥) = 𝐴))
139124, 130, 1383bitr2d 307 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 𝐴))
140123, 139orbi12d 919 . . . . . . 7 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
141120, 140mpbird 257 . . . . . 6 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))
142141ex 412 . . . . 5 ((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
143142ralrimivva 3202 . . . 4 (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
14425, 26, 2, 27isirred2 20421 . . . 4 (𝐴𝐼 ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ (Unit‘ℤring) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))))
14571, 84, 143, 144syl3anbrc 1344 . . 3 (𝐴 ∈ ℙ → 𝐴𝐼)
146145adantl 481 . 2 ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴𝐼)
14770, 146impbida 801 1 (𝐴 ∈ ℕ → (𝐴𝐼𝐴 ∈ ℙ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295  cle 11296   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  abscabs 15273  cdvds 16290  cprime 16708  Ringcrg 20230  Unitcui 20355  Irredcir 20356  ringczring 21457
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-prm 16709  df-gz 16968  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-irred 20359  df-invr 20388  df-dvr 20401  df-subrng 20546  df-subrg 20570  df-drng 20731  df-cnfld 21365  df-zring 21458
This theorem is referenced by:  dfprm2  21484  prmirred  21485
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