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Theorem prmirredlem 20057
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 20037 . . . . . 6 ring ∈ Ring
2 prmirred.i . . . . . . 7 𝐼 = (Irred‘ℤring)
3 zring1 20045 . . . . . . 7 1 = (1r‘ℤring)
42, 3irredn1 18915 . . . . . 6 ((ℤring ∈ Ring ∧ 𝐴𝐼) → 𝐴 ≠ 1)
51, 4mpan 664 . . . . 5 (𝐴𝐼𝐴 ≠ 1)
65anim2i 597 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7 eluz2b3 11966 . . . 4 (𝐴 ∈ (ℤ‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
86, 7sylibr 224 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ (ℤ‘2))
9 nnz 11602 . . . . . . . 8 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
109ad2antrl 701 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℤ)
11 simprr 750 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦𝐴)
12 nnne0 11256 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1312ad2antrl 701 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ≠ 0)
14 nnz 11602 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
1514ad2antrr 699 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℤ)
16 dvdsval2 15193 . . . . . . . . 9 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1710, 13, 15, 16syl3anc 1476 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1811, 17mpbid 222 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℤ)
1915zcnd 11686 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℂ)
20 nncn 11231 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2120ad2antrl 701 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℂ)
2219, 21, 13divcan2d 11006 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23 simplr 746 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴𝐼)
2422, 23eqeltrd 2850 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25 zringbas 20040 . . . . . . . 8 ℤ = (Base‘ℤring)
26 eqid 2771 . . . . . . . 8 (Unit‘ℤring) = (Unit‘ℤring)
27 zringmulr 20043 . . . . . . . 8 · = (.r‘ℤring)
282, 25, 26, 27irredmul 18918 . . . . . . 7 ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
2910, 18, 24, 28syl3anc 1476 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30 zringunit 20052 . . . . . . . . . 10 (𝑦 ∈ (Unit‘ℤring) ↔ (𝑦 ∈ ℤ ∧ (abs‘𝑦) = 1))
3130baib 519 . . . . . . . . 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 11502 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34 nn0re 11504 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
35 nn0ge0 11521 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
3634, 35absidd 14370 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
3733, 36syl 17 . . . . . . . . . 10 (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
3837ad2antrl 701 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘𝑦) = 𝑦)
3938eqeq1d 2773 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘𝑦) = 1 ↔ 𝑦 = 1))
4032, 39bitrd 268 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ 𝑦 = 1))
41 zringunit 20052 . . . . . . . . . 10 ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ ((𝐴 / 𝑦) ∈ ℤ ∧ (abs‘(𝐴 / 𝑦)) = 1))
4241baib 519 . . . . . . . . 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 11230 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
4544ad2antrr 699 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℝ)
46 simprl 748 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℕ)
4745, 46nndivred 11272 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48 nnnn0 11502 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49 nn0ge0 11521 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
5048, 49syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
5150ad2antrr 699 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ 𝐴)
5246nnred 11238 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℝ)
53 nngt0 11252 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ → 0 < 𝑦)
5453ad2antrl 701 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 < 𝑦)
55 divge0 11095 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
5645, 51, 52, 54, 55syl22anc 1477 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ (𝐴 / 𝑦))
5747, 56absidd 14370 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
5857eqeq1d 2773 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59 1cnd 10259 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 1 ∈ ℂ)
6019, 21, 59, 13divmuld 11026 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
6121mulid1d 10260 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · 1) = 𝑦)
6261eqeq1d 2773 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝑦 · 1) = 𝐴𝑦 = 𝐴))
6358, 60, 623bitrd 294 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ 𝑦 = 𝐴))
6443, 63bitrd 268 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ 𝑦 = 𝐴))
6540, 64orbi12d 896 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)) ↔ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6629, 65mpbid 222 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 = 1 ∨ 𝑦 = 𝐴))
6766expr 444 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ 𝑦 ∈ ℕ) → (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6867ralrimiva 3115 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69 isprm2 15603 . . 3 (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ‘2) ∧ ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
708, 68, 69sylanbrc 566 . 2 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ ℙ)
71 prmz 15597 . . . 4 (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72 1nprm 15600 . . . . 5 ¬ 1 ∈ ℙ
73 zringunit 20052 . . . . . 6 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74 prmnn 15596 . . . . . . . . . 10 (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75 nn0re 11504 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
7675, 49absidd 14370 . . . . . . . . . 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 2850 . . . . . . . 8 (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80 eleq1 2838 . . . . . . . 8 ((abs‘𝐴) = 1 → ((abs‘𝐴) ∈ ℙ ↔ 1 ∈ ℙ))
8179, 80syl5ibcom 235 . . . . . . 7 (𝐴 ∈ ℙ → ((abs‘𝐴) = 1 → 1 ∈ ℙ))
8281adantld 474 . . . . . 6 (𝐴 ∈ ℙ → ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℙ))
8373, 82syl5bi 232 . . . . 5 (𝐴 ∈ ℙ → (𝐴 ∈ (Unit‘ℤring) → 1 ∈ ℙ))
8472, 83mtoi 190 . . . 4 (𝐴 ∈ ℙ → ¬ 𝐴 ∈ (Unit‘ℤring))
85 dvdsmul1 15213 . . . . . . . . . . 11 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
8685ad2antlr 700 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ (𝑥 · 𝑦))
87 simpr 471 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) = 𝐴)
8886, 87breqtrd 4813 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥𝐴)
89 simplrl 756 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
9071ad2antrr 699 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
91 absdvdsb 15210 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
9289, 90, 91syl2anc 567 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
9388, 92mpbid 222 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∥ 𝐴)
94 breq1 4790 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → (𝑦𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95 eqeq1 2775 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96 eqeq1 2775 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
9795, 96orbi12d 896 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
9894, 97imbi12d 333 . . . . . . . . 9 (𝑦 = (abs‘𝑥) → ((𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
9969simprbi 480 . . . . . . . . . 10 (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10099ad2antrr 699 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10189zcnd 11686 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
10274ad2antrr 699 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
103102nnne0d 11268 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ≠ 0)
104 simplrr 757 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℤ)
105104zcnd 11686 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106105mul02d 10437 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107103, 87, 1063netr4d 3020 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108 oveq1 6801 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109108necon3i 2975 . . . . . . . . . . . . 13 ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
111101, 110absne0d 14395 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112111neneqd 2948 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113 nn0abscl 14261 . . . . . . . . . . . . 13 (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
11489, 113syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
115 elnn0 11497 . . . . . . . . . . . 12 ((abs‘𝑥) ∈ ℕ0 ↔ ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
116114, 115sylib 208 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
117116ord 845 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (¬ (abs‘𝑥) ∈ ℕ → (abs‘𝑥) = 0))
118112, 117mt3d 142 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ)
11998, 100, 118rspcdva 3467 . . . . . . . 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 20052 . . . . . . . . . 10 (𝑥 ∈ (Unit‘ℤring) ↔ (𝑥 ∈ ℤ ∧ (abs‘𝑥) = 1))
122121baib 519 . . . . . . . . 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 14384 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126125recnd 10271 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127 1cnd 10259 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128101abscld 14384 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129128recnd 10271 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130126, 127, 129, 111mulcand 10863 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
13187fveq2d 6337 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132101, 105absmuld 14402 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
13377ad2antrr 699 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134131, 132, 1333eqtr3d 2813 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135129mulid1d 10260 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136134, 135eqeq12d 2786 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137 eqcom 2778 . . . . . . . . . 10 (𝐴 = (abs‘𝑥) ↔ (abs‘𝑥) = 𝐴)
138136, 137syl6bb 276 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑥) = 𝐴))
139124, 130, 1383bitr2d 296 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 𝐴))
140123, 139orbi12d 896 . . . . . . 7 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
141120, 140mpbird 247 . . . . . 6 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))
142141ex 397 . . . . 5 ((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
143142ralrimivva 3120 . . . 4 (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
14425, 26, 2, 27isirred2 18910 . . . 4 (𝐴𝐼 ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ (Unit‘ℤring) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))))
14571, 84, 143, 144syl3anbrc 1428 . . 3 (𝐴 ∈ ℙ → 𝐴𝐼)
146145adantl 467 . 2 ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴𝐼)
14770, 146impbida 796 1 (𝐴 ∈ ℕ → (𝐴𝐼𝐴 ∈ ℙ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 828   = wceq 1631  wcel 2145  wne 2943  wral 3061   class class class wbr 4787  cfv 6032  (class class class)co 6794  cc 10137  cr 10138  0cc0 10139  1c1 10140   · cmul 10144   < clt 10277  cle 10278   / cdiv 10887  cn 11223  2c2 11273  0cn0 11495  cz 11580  cuz 11889  abscabs 14183  cdvds 15190  cprime 15593  Ringcrg 18756  Unitcui 18848  Irredcir 18849  ringzring 20034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216  ax-pre-sup 10217  ax-addf 10218  ax-mulf 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-tpos 7505  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-2o 7715  df-oadd 7718  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-sup 8505  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-div 10888  df-nn 11224  df-2 11282  df-3 11283  df-4 11284  df-5 11285  df-6 11286  df-7 11287  df-8 11288  df-9 11289  df-n0 11496  df-z 11581  df-dec 11697  df-uz 11890  df-rp 12037  df-fz 12535  df-seq 13010  df-exp 13069  df-cj 14048  df-re 14049  df-im 14050  df-sqrt 14184  df-abs 14185  df-dvds 15191  df-prm 15594  df-gz 15842  df-struct 16067  df-ndx 16068  df-slot 16069  df-base 16071  df-sets 16072  df-ress 16073  df-plusg 16163  df-mulr 16164  df-starv 16165  df-tset 16169  df-ple 16170  df-ds 16173  df-unif 16174  df-0g 16311  df-mgm 17451  df-sgrp 17493  df-mnd 17504  df-grp 17634  df-minusg 17635  df-subg 17800  df-cmn 18403  df-mgp 18699  df-ur 18711  df-ring 18758  df-cring 18759  df-oppr 18832  df-dvdsr 18850  df-unit 18851  df-irred 18852  df-invr 18881  df-dvr 18892  df-drng 18960  df-subrg 18989  df-cnfld 19963  df-zring 20035
This theorem is referenced by:  dfprm2  20058  prmirred  20059
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