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Theorem nrgring 24593
Description: A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nrgring (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)

Proof of Theorem nrgring
StepHypRef Expression
1 eqid 2728 . . 3 (norm‘𝑅) = (norm‘𝑅)
2 eqid 2728 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2nrgabv 24591 . 2 (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅))
42abvrcl 20701 . 2 ((norm‘𝑅) ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring)
53, 4syl 17 1 (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  cfv 6548  Ringcrg 20173  AbsValcabv 20696  normcnm 24498  NrmRingcnrg 24501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-abv 20697  df-nrg 24507
This theorem is referenced by:  nrgdsdi  24595  nrgdsdir  24596  nmdvr  24600  nrgtgp  24602  rlmnlm  24618  nrgtrg  24620  nrginvrcnlem  24621  nrginvrcn  24622  nrgtdrg  24623  rlmbn  25302  iistmd  33503  zrhnm  33570  cnzh  33571  rezh  33572
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