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

Theorem irredcl 19383
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 2818 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
3 irredn0.i . . 3 𝐼 = (Irred‘𝑅)
4 eqid 2818 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isirred2 19380 . 2 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋 ∈ (Unit‘𝑅) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑋 → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))))
65simp1bi 1137 1 (𝑋𝐼𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 841   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  Unitcui 19318  Irredcir 19319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-irred 19322
This theorem is referenced by:  irredrmul  19386  irredneg  19389  prmirred  20570
  Copyright terms: Public domain W3C validator