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Theorem prmirredlem 21500
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 21477 . . . . . 6 ring ∈ Ring
2 prmirred.i . . . . . . 7 𝐼 = (Irred‘ℤring)
3 zring1 21487 . . . . . . 7 1 = (1r‘ℤring)
42, 3irredn1 20442 . . . . . 6 ((ℤring ∈ Ring ∧ 𝐴𝐼) → 𝐴 ≠ 1)
51, 4mpan 690 . . . . 5 (𝐴𝐼𝐴 ≠ 1)
65anim2i 617 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7 eluz2b3 12961 . . . 4 (𝐴 ∈ (ℤ‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
86, 7sylibr 234 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ (ℤ‘2))
9 nnz 12631 . . . . . . . 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 12297 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1312ad2antrl 728 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ≠ 0)
14 nnz 12631 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
1514ad2antrr 726 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℤ)
16 dvdsval2 16289 . . . . . . . . 9 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1710, 13, 15, 16syl3anc 1370 . . . . . . . 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 12720 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℂ)
20 nncn 12271 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2120ad2antrl 728 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℂ)
2219, 21, 13divcan2d 12042 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23 simplr 769 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴𝐼)
2422, 23eqeltrd 2838 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25 zringbas 21481 . . . . . . . 8 ℤ = (Base‘ℤring)
26 eqid 2734 . . . . . . . 8 (Unit‘ℤring) = (Unit‘ℤring)
27 zringmulr 21485 . . . . . . . 8 · = (.r‘ℤring)
282, 25, 26, 27irredmul 20445 . . . . . . 7 ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
2910, 18, 24, 28syl3anc 1370 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30 zringunit 21494 . . . . . . . . . 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 12530 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34 nn0re 12532 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
35 nn0ge0 12548 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
3634, 35absidd 15457 . . . . . . . . . . 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 2736 . . . . . . . 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 21494 . . . . . . . . . 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 12270 . . . . . . . . . . . . 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 12317 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48 nnnn0 12530 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49 nn0ge0 12548 . . . . . . . . . . . . . 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 12278 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℝ)
53 nngt0 12294 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ → 0 < 𝑦)
5453ad2antrl 728 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 < 𝑦)
55 divge0 12134 . . . . . . . . . . . 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 15457 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
5857eqeq1d 2736 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59 1cnd 11253 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 1 ∈ ℂ)
6019, 21, 59, 13divmuld 12062 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
6121mulridd 11275 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · 1) = 𝑦)
6261eqeq1d 2736 . . . . . . . . 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 918 . . . . . 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 3143 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69 isprm2 16715 . . 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 16708 . . . 4 (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72 1nprm 16712 . . . . 5 ¬ 1 ∈ ℙ
73 zringunit 21494 . . . . . 6 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74 prmnn 16707 . . . . . . . . . 10 (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75 nn0re 12532 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
7675, 49absidd 15457 . . . . . . . . . 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 2838 . . . . . . . 8 (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80 eleq1 2826 . . . . . . . 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 16311 . . . . . . . . . . 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 5173 . . . . . . . . 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 16308 . . . . . . . . . 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 5150 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → (𝑦𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95 eqeq1 2738 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96 eqeq1 2738 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
9795, 96orbi12d 918 . . . . . . . . . 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 12720 . . . . . . . . . . . 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 12313 . . . . . . . . . . . . . 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 12720 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106105mul02d 11456 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107103, 87, 1063netr4d 3015 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108 oveq1 7437 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109108necon3i 2970 . . . . . . . . . . . . 13 ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
111101, 110absne0d 15482 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112111neneqd 2942 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113 nn0abscl 15347 . . . . . . . . . . . . 13 (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
11489, 113syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
115 elnn0 12525 . . . . . . . . . . . 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 864 . . . . . . . . . 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 3622 . . . . . . . 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 21494 . . . . . . . . . 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 15471 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126125recnd 11286 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127 1cnd 11253 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128101abscld 15471 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129128recnd 11286 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130126, 127, 129, 111mulcand 11893 . . . . . . . . 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 15489 . . . . . . . . . . . 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 2782 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135129mulridd 11275 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136134, 135eqeq12d 2750 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137 eqcom 2741 . . . . . . . . . 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 918 . . . . . . 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 3199 . . . 4 (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
14425, 26, 2, 27isirred2 20437 . . . 4 (𝐴𝐼 ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ (Unit‘ℤring) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))))
14571, 84, 143, 144syl3anbrc 1342 . . 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 847   = wceq 1536  wcel 2105  wne 2937  wral 3058   class class class wbr 5147  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   · cmul 11157   < clt 11292  cle 11293   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  abscabs 15269  cdvds 16286  cprime 16704  Ringcrg 20250  Unitcui 20371  Irredcir 20372  ringczring 21474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-rp 13032  df-fz 13544  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-prm 16705  df-gz 16963  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-subg 19153  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-irred 20375  df-invr 20404  df-dvr 20417  df-subrng 20562  df-subrg 20586  df-drng 20747  df-cnfld 21382  df-zring 21475
This theorem is referenced by:  dfprm2  21501  prmirred  21502
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