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Theorem nrgabv 18570
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 18569 . 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 1649    e. wcel 1717   ` cfv 5396  AbsValcabv 15833   normcnm 18497  NrmGrpcngp 18498  NrmRingcnrg 18500
This theorem is referenced by:  nrgrng  18572  nmmul  18573  nm1  18576  nrgdomn  18580  subrgnrg  18582  sranlm  18593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-nrg 18506
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