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Theorem prmirredlem 21042
Description: A positive integer is irreducible over β„€ 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 = (Irredβ€˜β„€ring)
Assertion
Ref Expression
prmirredlem (𝐴 ∈ β„• β†’ (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ β„™))

Proof of Theorem prmirredlem
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zringring 21020 . . . . . 6 β„€ring ∈ Ring
2 prmirred.i . . . . . . 7 𝐼 = (Irredβ€˜β„€ring)
3 zring1 21029 . . . . . . 7 1 = (1rβ€˜β„€ring)
42, 3irredn1 20240 . . . . . 6 ((β„€ring ∈ Ring ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 β‰  1)
51, 4mpan 689 . . . . 5 (𝐴 ∈ 𝐼 β†’ 𝐴 β‰  1)
65anim2i 618 . . . 4 ((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) β†’ (𝐴 ∈ β„• ∧ 𝐴 β‰  1))
7 eluz2b3 12906 . . . 4 (𝐴 ∈ (β„€β‰₯β€˜2) ↔ (𝐴 ∈ β„• ∧ 𝐴 β‰  1))
86, 7sylibr 233 . . 3 ((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 ∈ (β„€β‰₯β€˜2))
9 nnz 12579 . . . . . . . 8 (𝑦 ∈ β„• β†’ 𝑦 ∈ β„€)
109ad2antrl 727 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝑦 ∈ β„€)
11 simprr 772 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝑦 βˆ₯ 𝐴)
12 nnne0 12246 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 β‰  0)
1312ad2antrl 727 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝑦 β‰  0)
14 nnz 12579 . . . . . . . . . 10 (𝐴 ∈ β„• β†’ 𝐴 ∈ β„€)
1514ad2antrr 725 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝐴 ∈ β„€)
16 dvdsval2 16200 . . . . . . . . 9 ((𝑦 ∈ β„€ ∧ 𝑦 β‰  0 ∧ 𝐴 ∈ β„€) β†’ (𝑦 βˆ₯ 𝐴 ↔ (𝐴 / 𝑦) ∈ β„€))
1710, 13, 15, 16syl3anc 1372 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 βˆ₯ 𝐴 ↔ (𝐴 / 𝑦) ∈ β„€))
1811, 17mpbid 231 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝐴 / 𝑦) ∈ β„€)
1915zcnd 12667 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝐴 ∈ β„‚)
20 nncn 12220 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 ∈ β„‚)
2120ad2antrl 727 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝑦 ∈ β„‚)
2219, 21, 13divcan2d 11992 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 Β· (𝐴 / 𝑦)) = 𝐴)
23 simplr 768 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝐴 ∈ 𝐼)
2422, 23eqeltrd 2834 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 Β· (𝐴 / 𝑦)) ∈ 𝐼)
25 zringbas 21023 . . . . . . . 8 β„€ = (Baseβ€˜β„€ring)
26 eqid 2733 . . . . . . . 8 (Unitβ€˜β„€ring) = (Unitβ€˜β„€ring)
27 zringmulr 21027 . . . . . . . 8 Β· = (.rβ€˜β„€ring)
282, 25, 26, 27irredmul 20243 . . . . . . 7 ((𝑦 ∈ β„€ ∧ (𝐴 / 𝑦) ∈ β„€ ∧ (𝑦 Β· (𝐴 / 𝑦)) ∈ 𝐼) β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ∨ (𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring)))
2910, 18, 24, 28syl3anc 1372 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ∨ (𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring)))
30 zringunit 21036 . . . . . . . . . 10 (𝑦 ∈ (Unitβ€˜β„€ring) ↔ (𝑦 ∈ β„€ ∧ (absβ€˜π‘¦) = 1))
3130baib 537 . . . . . . . . 9 (𝑦 ∈ β„€ β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜π‘¦) = 1))
3210, 31syl 17 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜π‘¦) = 1))
33 nnnn0 12479 . . . . . . . . . . 11 (𝑦 ∈ β„• β†’ 𝑦 ∈ β„•0)
34 nn0re 12481 . . . . . . . . . . . 12 (𝑦 ∈ β„•0 β†’ 𝑦 ∈ ℝ)
35 nn0ge0 12497 . . . . . . . . . . . 12 (𝑦 ∈ β„•0 β†’ 0 ≀ 𝑦)
3634, 35absidd 15369 . . . . . . . . . . 11 (𝑦 ∈ β„•0 β†’ (absβ€˜π‘¦) = 𝑦)
3733, 36syl 17 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ (absβ€˜π‘¦) = 𝑦)
3837ad2antrl 727 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (absβ€˜π‘¦) = 𝑦)
3938eqeq1d 2735 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((absβ€˜π‘¦) = 1 ↔ 𝑦 = 1))
4032, 39bitrd 279 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ↔ 𝑦 = 1))
41 zringunit 21036 . . . . . . . . . 10 ((𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring) ↔ ((𝐴 / 𝑦) ∈ β„€ ∧ (absβ€˜(𝐴 / 𝑦)) = 1))
4241baib 537 . . . . . . . . 9 ((𝐴 / 𝑦) ∈ β„€ β†’ ((𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜(𝐴 / 𝑦)) = 1))
4318, 42syl 17 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜(𝐴 / 𝑦)) = 1))
44 nnre 12219 . . . . . . . . . . . . 13 (𝐴 ∈ β„• β†’ 𝐴 ∈ ℝ)
4544ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝐴 ∈ ℝ)
46 simprl 770 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝑦 ∈ β„•)
4745, 46nndivred 12266 . . . . . . . . . . 11 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝐴 / 𝑦) ∈ ℝ)
48 nnnn0 12479 . . . . . . . . . . . . . 14 (𝐴 ∈ β„• β†’ 𝐴 ∈ β„•0)
49 nn0ge0 12497 . . . . . . . . . . . . . 14 (𝐴 ∈ β„•0 β†’ 0 ≀ 𝐴)
5048, 49syl 17 . . . . . . . . . . . . 13 (𝐴 ∈ β„• β†’ 0 ≀ 𝐴)
5150ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 0 ≀ 𝐴)
5246nnred 12227 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 𝑦 ∈ ℝ)
53 nngt0 12243 . . . . . . . . . . . . 13 (𝑦 ∈ β„• β†’ 0 < 𝑦)
5453ad2antrl 727 . . . . . . . . . . . 12 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 0 < 𝑦)
55 divge0 12083 . . . . . . . . . . . 12 (((𝐴 ∈ ℝ ∧ 0 ≀ 𝐴) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) β†’ 0 ≀ (𝐴 / 𝑦))
5645, 51, 52, 54, 55syl22anc 838 . . . . . . . . . . 11 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 0 ≀ (𝐴 / 𝑦))
5747, 56absidd 15369 . . . . . . . . . 10 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (absβ€˜(𝐴 / 𝑦)) = (𝐴 / 𝑦))
5857eqeq1d 2735 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((absβ€˜(𝐴 / 𝑦)) = 1 ↔ (𝐴 / 𝑦) = 1))
59 1cnd 11209 . . . . . . . . . 10 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ 1 ∈ β„‚)
6019, 21, 59, 13divmuld 12012 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((𝐴 / 𝑦) = 1 ↔ (𝑦 Β· 1) = 𝐴))
6121mulridd 11231 . . . . . . . . . 10 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 Β· 1) = 𝑦)
6261eqeq1d 2735 . . . . . . . . 9 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((𝑦 Β· 1) = 𝐴 ↔ 𝑦 = 𝐴))
6358, 60, 623bitrd 305 . . . . . . . 8 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((absβ€˜(𝐴 / 𝑦)) = 1 ↔ 𝑦 = 𝐴))
6443, 63bitrd 279 . . . . . . 7 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring) ↔ 𝑦 = 𝐴))
6540, 64orbi12d 918 . . . . . 6 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ ((𝑦 ∈ (Unitβ€˜β„€ring) ∨ (𝐴 / 𝑦) ∈ (Unitβ€˜β„€ring)) ↔ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6629, 65mpbid 231 . . . . 5 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ (𝑦 ∈ β„• ∧ 𝑦 βˆ₯ 𝐴)) β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴))
6766expr 458 . . . 4 (((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) ∧ 𝑦 ∈ β„•) β†’ (𝑦 βˆ₯ 𝐴 β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
6867ralrimiva 3147 . . 3 ((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) β†’ βˆ€π‘¦ ∈ β„• (𝑦 βˆ₯ 𝐴 β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
69 isprm2 16619 . . 3 (𝐴 ∈ β„™ ↔ (𝐴 ∈ (β„€β‰₯β€˜2) ∧ βˆ€π‘¦ ∈ β„• (𝑦 βˆ₯ 𝐴 β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴))))
708, 68, 69sylanbrc 584 . 2 ((𝐴 ∈ β„• ∧ 𝐴 ∈ 𝐼) β†’ 𝐴 ∈ β„™)
71 prmz 16612 . . . 4 (𝐴 ∈ β„™ β†’ 𝐴 ∈ β„€)
72 1nprm 16616 . . . . 5 Β¬ 1 ∈ β„™
73 zringunit 21036 . . . . . 6 (𝐴 ∈ (Unitβ€˜β„€ring) ↔ (𝐴 ∈ β„€ ∧ (absβ€˜π΄) = 1))
74 prmnn 16611 . . . . . . . . . 10 (𝐴 ∈ β„™ β†’ 𝐴 ∈ β„•)
75 nn0re 12481 . . . . . . . . . . 11 (𝐴 ∈ β„•0 β†’ 𝐴 ∈ ℝ)
7675, 49absidd 15369 . . . . . . . . . 10 (𝐴 ∈ β„•0 β†’ (absβ€˜π΄) = 𝐴)
7774, 48, 763syl 18 . . . . . . . . 9 (𝐴 ∈ β„™ β†’ (absβ€˜π΄) = 𝐴)
78 id 22 . . . . . . . . 9 (𝐴 ∈ β„™ β†’ 𝐴 ∈ β„™)
7977, 78eqeltrd 2834 . . . . . . . 8 (𝐴 ∈ β„™ β†’ (absβ€˜π΄) ∈ β„™)
80 eleq1 2822 . . . . . . . 8 ((absβ€˜π΄) = 1 β†’ ((absβ€˜π΄) ∈ β„™ ↔ 1 ∈ β„™))
8179, 80syl5ibcom 244 . . . . . . 7 (𝐴 ∈ β„™ β†’ ((absβ€˜π΄) = 1 β†’ 1 ∈ β„™))
8281adantld 492 . . . . . 6 (𝐴 ∈ β„™ β†’ ((𝐴 ∈ β„€ ∧ (absβ€˜π΄) = 1) β†’ 1 ∈ β„™))
8373, 82biimtrid 241 . . . . 5 (𝐴 ∈ β„™ β†’ (𝐴 ∈ (Unitβ€˜β„€ring) β†’ 1 ∈ β„™))
8472, 83mtoi 198 . . . 4 (𝐴 ∈ β„™ β†’ Β¬ 𝐴 ∈ (Unitβ€˜β„€ring))
85 dvdsmul1 16221 . . . . . . . . . . 11 ((π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ π‘₯ βˆ₯ (π‘₯ Β· 𝑦))
8685ad2antlr 726 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ π‘₯ βˆ₯ (π‘₯ Β· 𝑦))
87 simpr 486 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (π‘₯ Β· 𝑦) = 𝐴)
8886, 87breqtrd 5175 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ π‘₯ βˆ₯ 𝐴)
89 simplrl 776 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ π‘₯ ∈ β„€)
9071ad2antrr 725 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ 𝐴 ∈ β„€)
91 absdvdsb 16218 . . . . . . . . . 10 ((π‘₯ ∈ β„€ ∧ 𝐴 ∈ β„€) β†’ (π‘₯ βˆ₯ 𝐴 ↔ (absβ€˜π‘₯) βˆ₯ 𝐴))
9289, 90, 91syl2anc 585 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (π‘₯ βˆ₯ 𝐴 ↔ (absβ€˜π‘₯) βˆ₯ 𝐴))
9388, 92mpbid 231 . . . . . . . 8 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘₯) βˆ₯ 𝐴)
94 breq1 5152 . . . . . . . . . 10 (𝑦 = (absβ€˜π‘₯) β†’ (𝑦 βˆ₯ 𝐴 ↔ (absβ€˜π‘₯) βˆ₯ 𝐴))
95 eqeq1 2737 . . . . . . . . . . 11 (𝑦 = (absβ€˜π‘₯) β†’ (𝑦 = 1 ↔ (absβ€˜π‘₯) = 1))
96 eqeq1 2737 . . . . . . . . . . 11 (𝑦 = (absβ€˜π‘₯) β†’ (𝑦 = 𝐴 ↔ (absβ€˜π‘₯) = 𝐴))
9795, 96orbi12d 918 . . . . . . . . . 10 (𝑦 = (absβ€˜π‘₯) β†’ ((𝑦 = 1 ∨ 𝑦 = 𝐴) ↔ ((absβ€˜π‘₯) = 1 ∨ (absβ€˜π‘₯) = 𝐴)))
9894, 97imbi12d 345 . . . . . . . . 9 (𝑦 = (absβ€˜π‘₯) β†’ ((𝑦 βˆ₯ 𝐴 β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴)) ↔ ((absβ€˜π‘₯) βˆ₯ 𝐴 β†’ ((absβ€˜π‘₯) = 1 ∨ (absβ€˜π‘₯) = 𝐴))))
9969simprbi 498 . . . . . . . . . 10 (𝐴 ∈ β„™ β†’ βˆ€π‘¦ ∈ β„• (𝑦 βˆ₯ 𝐴 β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10099ad2antrr 725 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ βˆ€π‘¦ ∈ β„• (𝑦 βˆ₯ 𝐴 β†’ (𝑦 = 1 ∨ 𝑦 = 𝐴)))
10189zcnd 12667 . . . . . . . . . . . 12 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ π‘₯ ∈ β„‚)
10274ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ 𝐴 ∈ β„•)
103102nnne0d 12262 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ 𝐴 β‰  0)
104 simplrr 777 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ 𝑦 ∈ β„€)
105104zcnd 12667 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ 𝑦 ∈ β„‚)
106105mul02d 11412 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (0 Β· 𝑦) = 0)
107103, 87, 1063netr4d 3019 . . . . . . . . . . . . 13 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (π‘₯ Β· 𝑦) β‰  (0 Β· 𝑦))
108 oveq1 7416 . . . . . . . . . . . . . 14 (π‘₯ = 0 β†’ (π‘₯ Β· 𝑦) = (0 Β· 𝑦))
109108necon3i 2974 . . . . . . . . . . . . 13 ((π‘₯ Β· 𝑦) β‰  (0 Β· 𝑦) β†’ π‘₯ β‰  0)
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ π‘₯ β‰  0)
111101, 110absne0d 15394 . . . . . . . . . . 11 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘₯) β‰  0)
112111neneqd 2946 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ Β¬ (absβ€˜π‘₯) = 0)
113 nn0abscl 15259 . . . . . . . . . . . . 13 (π‘₯ ∈ β„€ β†’ (absβ€˜π‘₯) ∈ β„•0)
11489, 113syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘₯) ∈ β„•0)
115 elnn0 12474 . . . . . . . . . . . 12 ((absβ€˜π‘₯) ∈ β„•0 ↔ ((absβ€˜π‘₯) ∈ β„• ∨ (absβ€˜π‘₯) = 0))
116114, 115sylib 217 . . . . . . . . . . 11 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ ((absβ€˜π‘₯) ∈ β„• ∨ (absβ€˜π‘₯) = 0))
117116ord 863 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (Β¬ (absβ€˜π‘₯) ∈ β„• β†’ (absβ€˜π‘₯) = 0))
118112, 117mt3d 148 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘₯) ∈ β„•)
11998, 100, 118rspcdva 3614 . . . . . . . 8 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ ((absβ€˜π‘₯) βˆ₯ 𝐴 β†’ ((absβ€˜π‘₯) = 1 ∨ (absβ€˜π‘₯) = 𝐴)))
12093, 119mpd 15 . . . . . . 7 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ ((absβ€˜π‘₯) = 1 ∨ (absβ€˜π‘₯) = 𝐴))
121 zringunit 21036 . . . . . . . . . 10 (π‘₯ ∈ (Unitβ€˜β„€ring) ↔ (π‘₯ ∈ β„€ ∧ (absβ€˜π‘₯) = 1))
122121baib 537 . . . . . . . . 9 (π‘₯ ∈ β„€ β†’ (π‘₯ ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜π‘₯) = 1))
12389, 122syl 17 . . . . . . . 8 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (π‘₯ ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜π‘₯) = 1))
124104, 31syl 17 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜π‘¦) = 1))
125105abscld 15383 . . . . . . . . . . 11 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘¦) ∈ ℝ)
126125recnd 11242 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘¦) ∈ β„‚)
127 1cnd 11209 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ 1 ∈ β„‚)
128101abscld 15383 . . . . . . . . . . 11 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘₯) ∈ ℝ)
129128recnd 11242 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π‘₯) ∈ β„‚)
130126, 127, 129, 111mulcand 11847 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (((absβ€˜π‘₯) Β· (absβ€˜π‘¦)) = ((absβ€˜π‘₯) Β· 1) ↔ (absβ€˜π‘¦) = 1))
13187fveq2d 6896 . . . . . . . . . . . 12 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜(π‘₯ Β· 𝑦)) = (absβ€˜π΄))
132101, 105absmuld 15401 . . . . . . . . . . . 12 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜(π‘₯ Β· 𝑦)) = ((absβ€˜π‘₯) Β· (absβ€˜π‘¦)))
13377ad2antrr 725 . . . . . . . . . . . 12 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (absβ€˜π΄) = 𝐴)
134131, 132, 1333eqtr3d 2781 . . . . . . . . . . 11 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ ((absβ€˜π‘₯) Β· (absβ€˜π‘¦)) = 𝐴)
135129mulridd 11231 . . . . . . . . . . 11 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ ((absβ€˜π‘₯) Β· 1) = (absβ€˜π‘₯))
136134, 135eqeq12d 2749 . . . . . . . . . 10 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (((absβ€˜π‘₯) Β· (absβ€˜π‘¦)) = ((absβ€˜π‘₯) Β· 1) ↔ 𝐴 = (absβ€˜π‘₯)))
137 eqcom 2740 . . . . . . . . . 10 (𝐴 = (absβ€˜π‘₯) ↔ (absβ€˜π‘₯) = 𝐴)
138136, 137bitrdi 287 . . . . . . . . 9 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (((absβ€˜π‘₯) Β· (absβ€˜π‘¦)) = ((absβ€˜π‘₯) Β· 1) ↔ (absβ€˜π‘₯) = 𝐴))
139124, 130, 1383bitr2d 307 . . . . . . . 8 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (𝑦 ∈ (Unitβ€˜β„€ring) ↔ (absβ€˜π‘₯) = 𝐴))
140123, 139orbi12d 918 . . . . . . 7 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ ((π‘₯ ∈ (Unitβ€˜β„€ring) ∨ 𝑦 ∈ (Unitβ€˜β„€ring)) ↔ ((absβ€˜π‘₯) = 1 ∨ (absβ€˜π‘₯) = 𝐴)))
141120, 140mpbird 257 . . . . . 6 (((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) ∧ (π‘₯ Β· 𝑦) = 𝐴) β†’ (π‘₯ ∈ (Unitβ€˜β„€ring) ∨ 𝑦 ∈ (Unitβ€˜β„€ring)))
142141ex 414 . . . . 5 ((𝐴 ∈ β„™ ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€)) β†’ ((π‘₯ Β· 𝑦) = 𝐴 β†’ (π‘₯ ∈ (Unitβ€˜β„€ring) ∨ 𝑦 ∈ (Unitβ€˜β„€ring))))
143142ralrimivva 3201 . . . 4 (𝐴 ∈ β„™ β†’ βˆ€π‘₯ ∈ β„€ βˆ€π‘¦ ∈ β„€ ((π‘₯ Β· 𝑦) = 𝐴 β†’ (π‘₯ ∈ (Unitβ€˜β„€ring) ∨ 𝑦 ∈ (Unitβ€˜β„€ring))))
14425, 26, 2, 27isirred2 20235 . . . 4 (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ β„€ ∧ Β¬ 𝐴 ∈ (Unitβ€˜β„€ring) ∧ βˆ€π‘₯ ∈ β„€ βˆ€π‘¦ ∈ β„€ ((π‘₯ Β· 𝑦) = 𝐴 β†’ (π‘₯ ∈ (Unitβ€˜β„€ring) ∨ 𝑦 ∈ (Unitβ€˜β„€ring)))))
14571, 84, 143, 144syl3anbrc 1344 . . 3 (𝐴 ∈ β„™ β†’ 𝐴 ∈ 𝐼)
146145adantl 483 . 2 ((𝐴 ∈ β„• ∧ 𝐴 ∈ β„™) β†’ 𝐴 ∈ 𝐼)
14770, 146impbida 800 1 (𝐴 ∈ β„• β†’ (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ β„™))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   Β· cmul 11115   < clt 11248   ≀ cle 11249   / cdiv 11871  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  abscabs 15181   βˆ₯ cdvds 16197  β„™cprime 16608  Ringcrg 20056  Unitcui 20169  Irredcir 20170  β„€ringczring 21017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-rp 12975  df-fz 13485  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-dvds 16198  df-prm 16609  df-gz 16863  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-subg 19003  df-cmn 19650  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-irred 20173  df-invr 20202  df-dvr 20215  df-subrg 20317  df-drng 20359  df-cnfld 20945  df-zring 21018
This theorem is referenced by:  dfprm2  21043  prmirred  21044
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