MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  irredcl Structured version   Visualization version   GIF version

Theorem irredcl 19453
Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i 𝐼 = (Irred‘𝑅)
irredcl.b 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
irredcl (𝑋𝐼𝑋𝐵)

Proof of Theorem irredcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredcl.b . . 3 𝐵 = (Base‘𝑅)
2 eqid 2801 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
3 irredn0.i . . 3 𝐼 = (Irred‘𝑅)
4 eqid 2801 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isirred2 19450 . 2 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋 ∈ (Unit‘𝑅) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑋 → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))))
65simp1bi 1142 1 (𝑋𝐼𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 844   = wceq 1538  wcel 2112  wral 3109  cfv 6328  (class class class)co 7139  Basecbs 16478  .rcmulr 16561  Unitcui 19388  Irredcir 19389
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-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-ral 3114  df-rex 3115  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-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-irred 19392
This theorem is referenced by:  irredrmul  19456  irredneg  19459  prmirred  20191
  Copyright terms: Public domain W3C validator