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Theorem nrgring 23244
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 2820 . . 3 (norm‘𝑅) = (norm‘𝑅)
2 eqid 2820 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2nrgabv 23242 . 2 (𝑅 ∈ NrmRing → (norm‘𝑅) ∈ (AbsVal‘𝑅))
42abvrcl 19564 . 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 2114  cfv 6327  Ringcrg 19272  AbsValcabv 19559  normcnm 23158  NrmRingcnrg 23161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fv 6335  df-abv 19560  df-nrg 23167
This theorem is referenced by:  nrgdsdi  23246  nrgdsdir  23247  nmdvr  23251  nrgtgp  23253  rlmnlm  23269  nrgtrg  23271  nrginvrcnlem  23272  nrginvrcn  23273  nrgtdrg  23274  rlmbn  23940  iistmd  31149  zrhnm  31214  cnzh  31215  rezh  31216
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