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Theorem nrgabv 24177
Description: The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnrg.1 𝑁 = (normβ€˜π‘…)
isnrg.2 𝐴 = (AbsValβ€˜π‘…)
Assertion
Ref Expression
nrgabv (𝑅 ∈ NrmRing β†’ 𝑁 ∈ 𝐴)

Proof of Theorem nrgabv
StepHypRef Expression
1 isnrg.1 . . 3 𝑁 = (normβ€˜π‘…)
2 isnrg.2 . . 3 𝐴 = (AbsValβ€˜π‘…)
31, 2isnrg 24176 . 2 (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
43simprbi 497 1 (𝑅 ∈ NrmRing β†’ 𝑁 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  AbsValcabv 20423  normcnm 24084  NrmGrpcngp 24085  NrmRingcnrg 24087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-nrg 24093
This theorem is referenced by:  nrgring  24179  nmmul  24180  nm1  24183  nrgdomn  24187  subrgnrg  24189  sranlm  24200
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