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Theorem isnrg 18696
Description: A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnrg.1  |-  N  =  ( norm `  R
)
isnrg.2  |-  A  =  (AbsVal `  R )
Assertion
Ref Expression
isnrg  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )

Proof of Theorem isnrg
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . . . 4  |-  ( r  =  R  ->  ( norm `  r )  =  ( norm `  R
) )
2 isnrg.1 . . . 4  |-  N  =  ( norm `  R
)
31, 2syl6eqr 2486 . . 3  |-  ( r  =  R  ->  ( norm `  r )  =  N )
4 fveq2 5728 . . . 4  |-  ( r  =  R  ->  (AbsVal `  r )  =  (AbsVal `  R ) )
5 isnrg.2 . . . 4  |-  A  =  (AbsVal `  R )
64, 5syl6eqr 2486 . . 3  |-  ( r  =  R  ->  (AbsVal `  r )  =  A )
73, 6eleq12d 2504 . 2  |-  ( r  =  R  ->  (
( norm `  r )  e.  (AbsVal `  r )  <->  N  e.  A ) )
8 df-nrg 18633 . 2  |- NrmRing  =  {
r  e. NrmGrp  |  ( norm `  r )  e.  (AbsVal `  r ) }
97, 8elrab2 3094 1  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  AbsValcabv 15904   normcnm 18624  NrmGrpcngp 18625  NrmRingcnrg 18627
This theorem is referenced by:  nrgabv  18697  nrgngp  18698  subrgnrg  18709  tngnrg  18710  cnnrg  18815  zhmnrg  24351
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-nrg 18633
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