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Theorem irredmul 17484
 Description: If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i Irred
irredmul.b
irredmul.u Unit
irredmul.t
Assertion
Ref Expression
irredmul

Proof of Theorem irredmul
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredmul.b . . . . 5
2 irredmul.u . . . . 5 Unit
3 irredn0.i . . . . 5 Irred
4 irredmul.t . . . . 5
51, 2, 3, 4isirred2 17476 . . . 4
65simp3bi 1013 . . 3
7 eqid 2457 . . . 4
8 oveq1 6303 . . . . . . 7
98eqeq1d 2459 . . . . . 6
10 eleq1 2529 . . . . . . 7
1110orbi1d 702 . . . . . 6
129, 11imbi12d 320 . . . . 5
13 oveq2 6304 . . . . . . 7
1413eqeq1d 2459 . . . . . 6
15 eleq1 2529 . . . . . . 7
1615orbi2d 701 . . . . . 6
1714, 16imbi12d 320 . . . . 5
1812, 17rspc2v 3219 . . . 4
197, 18mpii 43 . . 3
206, 19syl5 32 . 2
21203impia 1193 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 368   wa 369   w3a 973   wceq 1395   wcel 1819  wral 2807  cfv 5594  (class class class)co 6296  cbs 14643  cmulr 14712  Unitcui 17414  Irredcir 17415 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-irred 17418 This theorem is referenced by:  prmirredlem  18649  prmirredlemOLD  18652
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