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Theorem lnrring 38203
 Description: Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lnrring (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)

Proof of Theorem lnrring
StepHypRef Expression
1 islnr 38202 . 2 (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
21simplbi 478 1 (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2140  ‘cfv 6050  Ringcrg 18768  ringLModcrglmod 19392  LNoeMclnm 38166  LNoeRclnr 38200 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-iota 6013  df-fv 6058  df-lnr 38201 This theorem is referenced by:  lnr2i  38207  hbtlem6  38220  hbt  38221
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