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Theorem prmirredlem 19773
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 19753 . . . . . 6 ring ∈ Ring
2 prmirred.i . . . . . . 7 𝐼 = (Irred‘ℤring)
3 zring1 19761 . . . . . . 7 1 = (1r‘ℤring)
42, 3irredn1 18638 . . . . . 6 ((ℤring ∈ Ring ∧ 𝐴𝐼) → 𝐴 ≠ 1)
51, 4mpan 705 . . . . 5 (𝐴𝐼𝐴 ≠ 1)
65anim2i 592 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
7 eluz2b3 11714 . . . 4 (𝐴 ∈ (ℤ‘2) ↔ (𝐴 ∈ ℕ ∧ 𝐴 ≠ 1))
86, 7sylibr 224 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ (ℤ‘2))
9 nnz 11351 . . . . . . . 8 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
109ad2antrl 763 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℤ)
11 simprr 795 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦𝐴)
12 nnne0 11005 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1312ad2antrl 763 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ≠ 0)
14 nnz 11351 . . . . . . . . . 10 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
1514ad2antrr 761 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℤ)
16 dvdsval2 14921 . . . . . . . . 9 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ 𝐴 ∈ ℤ) → (𝑦𝐴 ↔ (𝐴 / 𝑦) ∈ ℤ))
1710, 13, 15, 16syl3anc 1323 . . . . . . . 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 11435 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℂ)
20 nncn 10980 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
2120ad2antrl 763 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℂ)
2219, 21, 13divcan2d 10755 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) = 𝐴)
23 simplr 791 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴𝐼)
2422, 23eqeltrd 2698 . . . . . . 7 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼)
25 zringbas 19756 . . . . . . . 8 ℤ = (Base‘ℤring)
26 eqid 2621 . . . . . . . 8 (Unit‘ℤring) = (Unit‘ℤring)
27 zringmulr 19759 . . . . . . . 8 · = (.r‘ℤring)
282, 25, 26, 27irredmul 18641 . . . . . . 7 ((𝑦 ∈ ℤ ∧ (𝐴 / 𝑦) ∈ ℤ ∧ (𝑦 · (𝐴 / 𝑦)) ∈ 𝐼) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
2910, 18, 24, 28syl3anc 1323 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 ∈ (Unit‘ℤring) ∨ (𝐴 / 𝑦) ∈ (Unit‘ℤring)))
30 zringunit 19768 . . . . . . . . . 10 (𝑦 ∈ (Unit‘ℤring) ↔ (𝑦 ∈ ℤ ∧ (abs‘𝑦) = 1))
3130baib 943 . . . . . . . . 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 11251 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
34 nn0re 11253 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0𝑦 ∈ ℝ)
35 nn0ge0 11270 . . . . . . . . . . . 12 (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
3634, 35absidd 14103 . . . . . . . . . . 11 (𝑦 ∈ ℕ0 → (abs‘𝑦) = 𝑦)
3733, 36syl 17 . . . . . . . . . 10 (𝑦 ∈ ℕ → (abs‘𝑦) = 𝑦)
3837ad2antrl 763 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘𝑦) = 𝑦)
3938eqeq1d 2623 . . . . . . . 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 19768 . . . . . . . . . 10 ((𝐴 / 𝑦) ∈ (Unit‘ℤring) ↔ ((𝐴 / 𝑦) ∈ ℤ ∧ (abs‘(𝐴 / 𝑦)) = 1))
4241baib 943 . . . . . . . . 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 10979 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
4544ad2antrr 761 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝐴 ∈ ℝ)
46 simprl 793 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℕ)
4745, 46nndivred 11021 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝐴 / 𝑦) ∈ ℝ)
48 nnnn0 11251 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
49 nn0ge0 11270 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ0 → 0 ≤ 𝐴)
5048, 49syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → 0 ≤ 𝐴)
5150ad2antrr 761 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ 𝐴)
5246nnred 10987 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 𝑦 ∈ ℝ)
53 nngt0 11001 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ → 0 < 𝑦)
5453ad2antrl 763 . . . . . . . . . . . 12 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 < 𝑦)
55 divge0 10844 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → 0 ≤ (𝐴 / 𝑦))
5645, 51, 52, 54, 55syl22anc 1324 . . . . . . . . . . 11 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 0 ≤ (𝐴 / 𝑦))
5747, 56absidd 14103 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (abs‘(𝐴 / 𝑦)) = (𝐴 / 𝑦))
5857eqeq1d 2623 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((abs‘(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59 1cnd 10008 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → 1 ∈ ℂ)
6019, 21, 59, 13divmuld 10775 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → ((𝐴 / 𝑦) = 1 ↔ (𝑦 · 1) = 𝐴))
6121mulid1d 10009 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ (𝑦 ∈ ℕ ∧ 𝑦𝐴)) → (𝑦 · 1) = 𝑦)
6261eqeq1d 2623 . . . . . . . . 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 745 . . . . . 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 642 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐴𝐼) ∧ 𝑦 ∈ ℕ) → (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6867ralrimiva 2961 . . 3 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69 isprm2 15330 . . 3 (𝐴 ∈ ℙ ↔ (𝐴 ∈ (ℤ‘2) ∧ ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴))))
708, 68, 69sylanbrc 697 . 2 ((𝐴 ∈ ℕ ∧ 𝐴𝐼) → 𝐴 ∈ ℙ)
71 prmz 15324 . . . 4 (𝐴 ∈ ℙ → 𝐴 ∈ ℤ)
72 1nprm 15327 . . . . 5 ¬ 1 ∈ ℙ
73 zringunit 19768 . . . . . 6 (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1))
74 prmnn 15323 . . . . . . . . . 10 (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
75 nn0re 11253 . . . . . . . . . . 11 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
7675, 49absidd 14103 . . . . . . . . . 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 2698 . . . . . . . 8 (𝐴 ∈ ℙ → (abs‘𝐴) ∈ ℙ)
80 eleq1 2686 . . . . . . . 8 ((abs‘𝐴) = 1 → ((abs‘𝐴) ∈ ℙ ↔ 1 ∈ ℙ))
8179, 80syl5ibcom 235 . . . . . . 7 (𝐴 ∈ ℙ → ((abs‘𝐴) = 1 → 1 ∈ ℙ))
8281adantld 483 . . . . . 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 simplrl 799 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℤ)
8685zcnd 11435 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∈ ℂ)
8774ad2antrr 761 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℕ)
8887nnne0d 11017 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ≠ 0)
89 simpr 477 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) = 𝐴)
90 simplrr 800 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℤ)
9190zcnd 11435 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑦 ∈ ℂ)
9291mul02d 10186 . . . . . . . . . . . . 13 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (0 · 𝑦) = 0)
9388, 89, 923netr4d 2867 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 · 𝑦) ≠ (0 · 𝑦))
94 oveq1 6617 . . . . . . . . . . . . 13 (𝑥 = 0 → (𝑥 · 𝑦) = (0 · 𝑦))
9594necon3i 2822 . . . . . . . . . . . 12 ((𝑥 · 𝑦) ≠ (0 · 𝑦) → 𝑥 ≠ 0)
9693, 95syl 17 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ≠ 0)
9786, 96absne0d 14128 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ≠ 0)
9897neneqd 2795 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ¬ (abs‘𝑥) = 0)
99 nn0abscl 13994 . . . . . . . . . . . 12 (𝑥 ∈ ℤ → (abs‘𝑥) ∈ ℕ0)
10085, 99syl 17 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ0)
101 elnn0 11246 . . . . . . . . . . 11 ((abs‘𝑥) ∈ ℕ0 ↔ ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
102100, 101sylib 208 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) ∈ ℕ ∨ (abs‘𝑥) = 0))
103102ord 392 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (¬ (abs‘𝑥) ∈ ℕ → (abs‘𝑥) = 0))
10498, 103mt3d 140 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℕ)
10569simprbi 480 . . . . . . . . 9 (𝐴 ∈ ℙ → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
106105ad2antrr 761 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)))
107 dvdsmul1 14938 . . . . . . . . . . 11 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∥ (𝑥 · 𝑦))
108107ad2antlr 762 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥 ∥ (𝑥 · 𝑦))
109108, 89breqtrd 4644 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝑥𝐴)
11071ad2antrr 761 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 𝐴 ∈ ℤ)
111 absdvdsb 14935 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
11285, 110, 111syl2anc 692 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
113109, 112mpbid 222 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∥ 𝐴)
114 breq1 4621 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → (𝑦𝐴 ↔ (abs‘𝑥) ∥ 𝐴))
115 eqeq1 2625 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 1 ↔ (abs‘𝑥) = 1))
116 eqeq1 2625 . . . . . . . . . . 11 (𝑦 = (abs‘𝑥) → (𝑦 = 𝐴 ↔ (abs‘𝑥) = 𝐴))
117115, 116orbi12d 745 . . . . . . . . . 10 (𝑦 = (abs‘𝑥) → ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴)))
118114, 117imbi12d 334 . . . . . . . . 9 (𝑦 = (abs‘𝑥) → ((𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
119118rspcv 3294 . . . . . . . 8 ((abs‘𝑥) ∈ ℕ → (∀𝑦 ∈ ℕ (𝑦𝐴 → (𝑦 = 1 ∨ 𝑦 = 𝐴)) → ((abs‘𝑥) ∥ 𝐴 → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))))
120104, 106, 113, 119syl3c 66 . . . . . . 7 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) = 1 ∨ (abs‘𝑥) = 𝐴))
121 zringunit 19768 . . . . . . . . . 10 (𝑥 ∈ (Unit‘ℤring) ↔ (𝑥 ∈ ℤ ∧ (abs‘𝑥) = 1))
122121baib 943 . . . . . . . . 9 (𝑥 ∈ ℤ → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
12385, 122syl 17 . . . . . . . 8 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑥 ∈ (Unit‘ℤring) ↔ (abs‘𝑥) = 1))
12490, 31syl 17 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (𝑦 ∈ (Unit‘ℤring) ↔ (abs‘𝑦) = 1))
12591abscld 14117 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℝ)
126125recnd 10020 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑦) ∈ ℂ)
127 1cnd 10008 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → 1 ∈ ℂ)
12886abscld 14117 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℝ)
129128recnd 10020 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝑥) ∈ ℂ)
130126, 127, 129, 97mulcand 10612 . . . . . . . . 9 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ (abs‘𝑦) = 1))
13189fveq2d 6157 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = (abs‘𝐴))
13286, 91absmuld 14135 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦)))
13377ad2antrr 761 . . . . . . . . . . . 12 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (abs‘𝐴) = 𝐴)
134131, 132, 1333eqtr3d 2663 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · (abs‘𝑦)) = 𝐴)
135129mulid1d 10009 . . . . . . . . . . 11 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → ((abs‘𝑥) · 1) = (abs‘𝑥))
136134, 135eqeq12d 2636 . . . . . . . . . 10 (((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) ∧ (𝑥 · 𝑦) = 𝐴) → (((abs‘𝑥) · (abs‘𝑦)) = ((abs‘𝑥) · 1) ↔ 𝐴 = (abs‘𝑥)))
137 eqcom 2628 . . . . . . . . . 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 745 . . . . . . 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 450 . . . . 5 ((𝐴 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
143142ralrimivva 2966 . . . 4 (𝐴 ∈ ℙ → ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring))))
14425, 26, 2, 27isirred2 18633 . . . 4 (𝐴𝐼 ↔ (𝐴 ∈ ℤ ∧ ¬ 𝐴 ∈ (Unit‘ℤring) ∧ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ ((𝑥 · 𝑦) = 𝐴 → (𝑥 ∈ (Unit‘ℤring) ∨ 𝑦 ∈ (Unit‘ℤring)))))
14571, 84, 143, 144syl3anbrc 1244 . . 3 (𝐴 ∈ ℙ → 𝐴𝐼)
146145adantl 482 . 2 ((𝐴 ∈ ℕ ∧ 𝐴 ∈ ℙ) → 𝐴𝐼)
14770, 146impbida 876 1 (𝐴 ∈ ℕ → (𝐴𝐼𝐴 ∈ ℙ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907   class class class wbr 4618  cfv 5852  (class class class)co 6610  cc 9886  cr 9887  0cc0 9888  1c1 9889   · cmul 9893   < clt 10026  cle 10027   / cdiv 10636  cn 10972  2c2 11022  0cn0 11244  cz 11329  cuz 11639  abscabs 13916  cdvds 14918  cprime 15320  Ringcrg 18479  Unitcui 18571  Irredcir 18572  ringzring 19750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966  ax-addf 9967  ax-mulf 9968
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-rp 11785  df-fz 12277  df-seq 12750  df-exp 12809  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-dvds 14919  df-prm 15321  df-gz 15569  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-minusg 17358  df-subg 17523  df-cmn 18127  df-mgp 18422  df-ur 18434  df-ring 18481  df-cring 18482  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-irred 18575  df-invr 18604  df-dvr 18615  df-drng 18681  df-subrg 18710  df-cnfld 19679  df-zring 19751
This theorem is referenced by:  dfprm2  19774  prmirred  19775
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