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Theorem nrgring 24530
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 2726 . . 3 (normβ€˜π‘…) = (normβ€˜π‘…)
2 eqid 2726 . . 3 (AbsValβ€˜π‘…) = (AbsValβ€˜π‘…)
31, 2nrgabv 24528 . 2 (𝑅 ∈ NrmRing β†’ (normβ€˜π‘…) ∈ (AbsValβ€˜π‘…))
42abvrcl 20661 . 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 6536  Ringcrg 20135  AbsValcabv 20656  normcnm 24435  NrmRingcnrg 24438
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fv 6544  df-abv 20657  df-nrg 24444
This theorem is referenced by:  nrgdsdi  24532  nrgdsdir  24533  nmdvr  24537  nrgtgp  24539  rlmnlm  24555  nrgtrg  24557  nrginvrcnlem  24558  nrginvrcn  24559  nrgtdrg  24560  rlmbn  25239  iistmd  33411  zrhnm  33478  cnzh  33479  rezh  33480
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