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Theorem nrgabv 22275
 Description: The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnrg.1 𝑁 = (norm‘𝑅)
isnrg.2 𝐴 = (AbsVal‘𝑅)
Assertion
Ref Expression
nrgabv (𝑅 ∈ NrmRing → 𝑁𝐴)

Proof of Theorem nrgabv
StepHypRef Expression
1 isnrg.1 . . 3 𝑁 = (norm‘𝑅)
2 isnrg.2 . . 3 𝐴 = (AbsVal‘𝑅)
31, 2isnrg 22274 . 2 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))
43simprbi 479 1 (𝑅 ∈ NrmRing → 𝑁𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  AbsValcabv 18639  normcnm 22191  NrmGrpcngp 22192  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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  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-iota 5768  df-fv 5812  df-nrg 22200 This theorem is referenced by:  nrgring  22277  nmmul  22278  nm1  22281  nrgdomn  22285  subrgnrg  22287  sranlm  22298
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