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Theorem nrgring 24600
Description: A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nrgring (𝑅 ∈ NrmRing β†’ 𝑅 ∈ Ring)

Proof of Theorem nrgring
StepHypRef Expression
1 eqid 2728 . . 3 (normβ€˜π‘…) = (normβ€˜π‘…)
2 eqid 2728 . . 3 (AbsValβ€˜π‘…) = (AbsValβ€˜π‘…)
31, 2nrgabv 24598 . 2 (𝑅 ∈ NrmRing β†’ (normβ€˜π‘…) ∈ (AbsValβ€˜π‘…))
42abvrcl 20708 . 2 ((normβ€˜π‘…) ∈ (AbsValβ€˜π‘…) β†’ 𝑅 ∈ Ring)
53, 4syl 17 1 (𝑅 ∈ NrmRing β†’ 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  β€˜cfv 6553  Ringcrg 20180  AbsValcabv 20703  normcnm 24505  NrmRingcnrg 24508
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fv 6561  df-abv 20704  df-nrg 24514
This theorem is referenced by:  nrgdsdi  24602  nrgdsdir  24603  nmdvr  24607  nrgtgp  24609  rlmnlm  24625  nrgtrg  24627  nrginvrcnlem  24628  nrginvrcn  24629  nrgtdrg  24630  rlmbn  25309  iistmd  33536  zrhnm  33603  cnzh  33604  rezh  33605
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