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Theorem nrgabv 18687
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  e. NrmRing  ->  N  e.  A
)

Proof of Theorem nrgabv
StepHypRef Expression
1 isnrg.1 . . 3  |-  N  =  ( norm `  R
)
2 isnrg.2 . . 3  |-  A  =  (AbsVal `  R )
31, 2isnrg 18686 . 2  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )
43simprbi 451 1  |-  ( R  e. NrmRing  ->  N  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  AbsValcabv 15894   normcnm 18614  NrmGrpcngp 18615  NrmRingcnrg 18617
This theorem is referenced by:  nrgrng  18689  nmmul  18690  nm1  18693  nrgdomn  18697  subrgnrg  18699  sranlm  18710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-nrg 18623
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