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Theorem prmirredlem 20189
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 20169 . . . . . 6 ring ∈ Ring
2 prmirred.i . . . . . . 7 𝐼 = (Irred‘ℤring)
3 zring1 20177 . . . . . . 7 1 = (1r‘ℤring)
42, 3irredn1 19455 . . . . . 6 ((ℤring ∈ Ring ∧ 𝐴𝐼) → 𝐴 ≠ 1)
51, 4mpan 689 . . . . 5 (𝐴𝐼𝐴 ≠ 1)
65anim2i 619 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7 eluz2b3 12314 . . . 4 (𝐴 ∈ (ℤ‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
86, 7sylibr 237 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ (ℤ‘2))
9 nnz 11996 . . . . . . . 8 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
109ad2antrl 727 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℤ)
11 simprr 772 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦𝐴)
12 nnne0 11663 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1312ad2antrl 727 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ≠ 0)
14 nnz 11996 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
1514ad2antrr 725 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℤ)
16 dvdsval2 15605 . . . . . . . . 9 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1710, 13, 15, 16syl3anc 1368 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1811, 17mpbid 235 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℤ)
1915zcnd 12080 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℂ)
20 nncn 11637 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2120ad2antrl 727 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℂ)
2219, 21, 13divcan2d 11411 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23 simplr 768 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴𝐼)
2422, 23eqeltrd 2893 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25 zringbas 20172 . . . . . . . 8 ℤ = (Base‘ℤring)
26 eqid 2801 . . . . . . . 8 (Unit‘ℤring) = (Unit‘ℤring)
27 zringmulr 20175 . . . . . . . 8 · = (.r‘ℤring)
282, 25, 26, 27irredmul 19458 . . . . . . 7 ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
2910, 18, 24, 28syl3anc 1368 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30 zringunit 20184 . . . . . . . . . 10 (𝑦 ∈ (Unit‘ℤring) ↔ (𝑦 ∈ ℤ ∧ (abs‘𝑦) = 1))
3130baib 539 . . . . . . . . 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 11896 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34 nn0re 11898 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
35 nn0ge0 11914 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
3634, 35absidd 14777 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
3733, 36syl 17 . . . . . . . . . 10 (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
3837ad2antrl 727 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘𝑦) = 𝑦)
3938eqeq1d 2803 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘𝑦) = 1 ↔ 𝑦 = 1))
4032, 39bitrd 282 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ↔ 𝑦 = 1))
41 zringunit 20184 . . . . . . . . . 10 ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ ((𝐴 / 𝑦) ∈ ℤ ∧ (abs‘(𝐴 / 𝑦)) = 1))
4241baib 539 . . . . . . . . 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 11636 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
4544ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℝ)
46 simprl 770 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℕ)
4745, 46nndivred 11683 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48 nnnn0 11896 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49 nn0ge0 11914 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
5048, 49syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
5150ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ 𝐴)
5246nnred 11644 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℝ)
53 nngt0 11660 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ → 0 < 𝑦)
5453ad2antrl 727 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 < 𝑦)
55 divge0 11502 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
5645, 51, 52, 54, 55syl22anc 837 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ (𝐴 / 𝑦))
5747, 56absidd 14777 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
5857eqeq1d 2803 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59 1cnd 10629 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 1 ∈ ℂ)
6019, 21, 59, 13divmuld 11431 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
6121mulid1d 10651 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · 1) = 𝑦)
6261eqeq1d 2803 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝑦 · 1) = 𝐴𝑦 = 𝐴))
6358, 60, 623bitrd 308 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ 𝑦 = 𝐴))
6443, 63bitrd 282 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ 𝑦 = 𝐴))
6540, 64orbi12d 916 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)) ↔ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6629, 65mpbid 235 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 = 1 ∨ 𝑦 = 𝐴))
6766expr 460 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ 𝑦 ∈ ℕ) → (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6867ralrimiva 3152 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69 isprm2 16019 . . 3 (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ‘2) ∧ ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
708, 68, 69sylanbrc 586 . 2 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ ℙ)
71 prmz 16012 . . . 4 (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72 1nprm 16016 . . . . 5 ¬ 1 ∈ ℙ
73 zringunit 20184 . . . . . 6 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74 prmnn 16011 . . . . . . . . . 10 (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75 nn0re 11898 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
7675, 49absidd 14777 . . . . . . . . . 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 2893 . . . . . . . 8 (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80 eleq1 2880 . . . . . . . 8 ((abs‘𝐴) = 1 → ((abs‘𝐴) ∈ ℙ ↔ 1 ∈ ℙ))
8179, 80syl5ibcom 248 . . . . . . 7 (𝐴 ∈ ℙ → ((abs‘𝐴) = 1 → 1 ∈ ℙ))
8281adantld 494 . . . . . 6 (𝐴 ∈ ℙ → ((𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1) → 1 ∈ ℙ))
8373, 82syl5bi 245 . . . . 5 (𝐴 ∈ ℙ → (𝐴 ∈ (Unit‘ℤring) → 1 ∈ ℙ))
8472, 83mtoi 202 . . . 4 (𝐴 ∈ ℙ → ¬ 𝐴 ∈ (Unit‘ℤring))
85 dvdsmul1 15626 . . . . . . . . . . 11 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
8685ad2antlr 726 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ (𝑥 · 𝑦))
87 simpr 488 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) = 𝐴)
8886, 87breqtrd 5059 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥𝐴)
89 simplrl 776 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
9071ad2antrr 725 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
91 absdvdsb 15623 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
9289, 90, 91syl2anc 587 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
9388, 92mpbid 235 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∥ 𝐴)
94 breq1 5036 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → (𝑦𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
95 eqeq1 2805 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
96 eqeq1 2805 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
9795, 96orbi12d 916 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
9894, 97imbi12d 348 . . . . . . . . 9 (𝑦 = (abs‘𝑥) → ((𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
9969simprbi 500 . . . . . . . . . 10 (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10099ad2antrr 725 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10189zcnd 12080 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
10274ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
103102nnne0d 11679 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ≠ 0)
104 simplrr 777 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℤ)
105104zcnd 12080 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
106105mul02d 10831 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
107103, 87, 1063netr4d 3067 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
108 oveq1 7146 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
109108necon3i 3022 . . . . . . . . . . . . 13 ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
111101, 110absne0d 14802 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
112111neneqd 2995 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
113 nn0abscl 14667 . . . . . . . . . . . . 13 (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
11489, 113syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
115 elnn0 11891 . . . . . . . . . . . 12 ((abs‘𝑥) ∈ ℕ0 ↔ ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
116114, 115sylib 221 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
117116ord 861 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (¬ (abs‘𝑥) ∈ ℕ → (abs‘𝑥) = 0))
118112, 117mt3d 150 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ)
11998, 100, 118rspcdva 3576 . . . . . . . 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 20184 . . . . . . . . . 10 (𝑥 ∈ (Unit‘ℤring) ↔ (𝑥 ∈ ℤ ∧ (abs‘𝑥) = 1))
122121baib 539 . . . . . . . . 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 14791 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126125recnd 10662 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127 1cnd 10629 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
128101abscld 14791 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129128recnd 10662 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130126, 127, 129, 111mulcand 11266 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
13187fveq2d 6653 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
132101, 105absmuld 14809 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
13377ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134131, 132, 1333eqtr3d 2844 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135129mulid1d 10651 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136134, 135eqeq12d 2817 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137 eqcom 2808 . . . . . . . . . 10 (𝐴 = (abs‘𝑥) ↔ (abs‘𝑥) = 𝐴)
138136, 137syl6bb 290 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑥) = 𝐴))
139124, 130, 1383bitr2d 310 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 𝐴))
140123, 139orbi12d 916 . . . . . . 7 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
141120, 140mpbird 260 . . . . . 6 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))
142141ex 416 . . . . 5 ((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
143142ralrimivva 3159 . . . 4 (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
14425, 26, 2, 27isirred2 19450 . . . 4 (𝐴𝐼 ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ (Unit‘ℤring) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))))
14571, 84, 143, 144syl3anbrc 1340 . . 3 (𝐴 ∈ ℙ → 𝐴𝐼)
146145adantl 485 . 2 ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴𝐼)
14770, 146impbida 800 1 (𝐴 ∈ ℕ → (𝐴𝐼𝐴 ∈ ℙ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wral 3109   class class class wbr 5033  cfv 6328  (class class class)co 7139  cc 10528  cr 10529  0cc0 10530  1c1 10531   · cmul 10535   < clt 10668  cle 10669   / cdiv 11290  cn 11629  2c2 11684  0cn0 11889  cz 11973  cuz 12235  abscabs 14588  cdvds 15602  cprime 16008  Ringcrg 19293  Unitcui 19388  Irredcir 19389  ringzring 20166
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-rp 12382  df-fz 12890  df-seq 13369  df-exp 13430  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-dvds 15603  df-prm 16009  df-gz 16259  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-minusg 18102  df-subg 18271  df-cmn 18903  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-oppr 19372  df-dvdsr 19390  df-unit 19391  df-irred 19392  df-invr 19421  df-dvr 19432  df-drng 19500  df-subrg 19529  df-cnfld 20095  df-zring 20167
This theorem is referenced by:  dfprm2  20190  prmirred  20191
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