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Theorem nrgring 22277
 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 2610 . . 3 (norm‘𝑅) = (norm‘𝑅)
2 eqid 2610 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2nrgabv 22275 . 2 (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅))
42abvrcl 18644 . 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 1977  ‘cfv 5804  Ringcrg 18370  AbsValcabv 18639  normcnm 22191  NrmRingcnrg 22194 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-abv 18640  df-nrg 22200 This theorem is referenced by:  nrgdsdi  22279  nrgdsdir  22280  nmdvr  22284  nrgtgp  22286  rlmnlm  22302  nrgtrg  22304  nrginvrcnlem  22305  nrginvrcn  22306  nrgtdrg  22307  rlmbn  22965  iistmd  29276  zrhnm  29341  cnzh  29342  rezh  29343
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