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Theorem isnrg 18171
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 5525 . . . 4  |-  ( r  =  R  ->  ( norm `  r )  =  ( norm `  R
) )
2 isnrg.1 . . . 4  |-  N  =  ( norm `  R
)
31, 2syl6eqr 2333 . . 3  |-  ( r  =  R  ->  ( norm `  r )  =  N )
4 fveq2 5525 . . . 4  |-  ( r  =  R  ->  (AbsVal `  r )  =  (AbsVal `  R ) )
5 isnrg.2 . . . 4  |-  A  =  (AbsVal `  R )
64, 5syl6eqr 2333 . . 3  |-  ( r  =  R  ->  (AbsVal `  r )  =  A )
73, 6eleq12d 2351 . 2  |-  ( r  =  R  ->  (
( norm `  r )  e.  (AbsVal `  r )  <->  N  e.  A ) )
8 df-nrg 18108 . 2  |- NrmRing  =  {
r  e. NrmGrp  |  ( norm `  r )  e.  (AbsVal `  r ) }
97, 8elrab2 2925 1  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  AbsValcabv 15581   normcnm 18099  NrmGrpcngp 18100  NrmRingcnrg 18102
This theorem is referenced by:  nrgabv  18172  nrgngp  18173  subrgnrg  18184  tngnrg  18185  cnnrg  18290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-nrg 18108
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