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Theorem irredcl 20237
Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i 𝐼 = (Irredβ€˜π‘…)
irredcl.b 𝐡 = (Baseβ€˜π‘…)
Assertion
Ref Expression
irredcl (𝑋 ∈ 𝐼 β†’ 𝑋 ∈ 𝐡)

Proof of Theorem irredcl
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 irredcl.b . . 3 𝐡 = (Baseβ€˜π‘…)
2 eqid 2732 . . 3 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
3 irredn0.i . . 3 𝐼 = (Irredβ€˜π‘…)
4 eqid 2732 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
51, 2, 3, 4isirred2 20234 . 2 (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ∈ (Unitβ€˜π‘…) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯(.rβ€˜π‘…)𝑦) = 𝑋 β†’ (π‘₯ ∈ (Unitβ€˜π‘…) ∨ 𝑦 ∈ (Unitβ€˜π‘…)))))
65simp1bi 1145 1 (𝑋 ∈ 𝐼 β†’ 𝑋 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∨ wo 845   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  .rcmulr 17197  Unitcui 20168  Irredcir 20169
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-irred 20172
This theorem is referenced by:  irredrmul  20240  irredneg  20243  prmirred  21043
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