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Theorem irredcl 20134
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 2737 . . 3 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
3 irredn0.i . . 3 𝐼 = (Irredβ€˜π‘…)
4 eqid 2737 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
51, 2, 3, 4isirred2 20131 . 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 1146 1 (𝑋 ∈ 𝐼 β†’ 𝑋 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  .rcmulr 17135  Unitcui 20069  Irredcir 20070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-irred 20073
This theorem is referenced by:  irredrmul  20137  irredneg  20140  prmirred  20898
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