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Theorem irredcl 18473
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 2609 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
3 irredn0.i . . 3 𝐼 = (Irred‘𝑅)
4 eqid 2609 . . 3 (.r𝑅) = (.r𝑅)
51, 2, 3, 4isirred2 18470 . 2 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋 ∈ (Unit‘𝑅) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥(.r𝑅)𝑦) = 𝑋 → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))))
65simp1bi 1068 1 (𝑋𝐼𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381   = wceq 1474  wcel 1976  wral 2895  cfv 5790  (class class class)co 6527  Basecbs 15641  .rcmulr 15715  Unitcui 18408  Irredcir 18409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-irred 18412
This theorem is referenced by:  irredrmul  18476  irredneg  18479  prmirred  19607
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