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Theorem nrgring 22795
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 2799 . . 3 (norm‘𝑅) = (norm‘𝑅)
2 eqid 2799 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2nrgabv 22793 . 2 (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅))
42abvrcl 19139 . 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 2157  cfv 6101  Ringcrg 18863  AbsValcabv 19134  normcnm 22709  NrmRingcnrg 22712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fv 6109  df-abv 19135  df-nrg 22718
This theorem is referenced by:  nrgdsdi  22797  nrgdsdir  22798  nmdvr  22802  nrgtgp  22804  rlmnlm  22820  nrgtrg  22822  nrginvrcnlem  22823  nrginvrcn  22824  nrgtdrg  22825  rlmbn  23487  iistmd  30464  zrhnm  30529  cnzh  30530  rezh  30531
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