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Theorem irrednu 20369
Description: An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irredn0.i 𝐼 = (Irredβ€˜π‘…)
irrednu.u π‘ˆ = (Unitβ€˜π‘…)
Assertion
Ref Expression
irrednu (𝑋 ∈ 𝐼 β†’ Β¬ 𝑋 ∈ π‘ˆ)

Proof of Theorem irrednu
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2727 . . 3 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2 irrednu.u . . 3 π‘ˆ = (Unitβ€˜π‘…)
3 irredn0.i . . 3 𝐼 = (Irredβ€˜π‘…)
4 eqid 2727 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
51, 2, 3, 4isirred2 20365 . 2 (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (Baseβ€˜π‘…) ∧ Β¬ 𝑋 ∈ π‘ˆ ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘…)βˆ€π‘¦ ∈ (Baseβ€˜π‘…)((π‘₯(.rβ€˜π‘…)𝑦) = 𝑋 β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ))))
65simp2bi 1143 1 (𝑋 ∈ 𝐼 β†’ Β¬ 𝑋 ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∨ wo 845   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  .rcmulr 17239  Unitcui 20299  Irredcir 20300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-irred 20303
This theorem is referenced by:  irredn1  20370  mxidlirred  33203  irredminply  33389
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