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Theorem isnrg 20239
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 5689 . . . 4  |-  ( r  =  R  ->  ( norm `  r )  =  ( norm `  R
) )
2 isnrg.1 . . . 4  |-  N  =  ( norm `  R
)
31, 2syl6eqr 2491 . . 3  |-  ( r  =  R  ->  ( norm `  r )  =  N )
4 fveq2 5689 . . . 4  |-  ( r  =  R  ->  (AbsVal `  r )  =  (AbsVal `  R ) )
5 isnrg.2 . . . 4  |-  A  =  (AbsVal `  R )
64, 5syl6eqr 2491 . . 3  |-  ( r  =  R  ->  (AbsVal `  r )  =  A )
73, 6eleq12d 2509 . 2  |-  ( r  =  R  ->  (
( norm `  r )  e.  (AbsVal `  r )  <->  N  e.  A ) )
8 df-nrg 20176 . 2  |- NrmRing  =  {
r  e. NrmGrp  |  ( norm `  r )  e.  (AbsVal `  r ) }
97, 8elrab2 3117 1  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5416  AbsValcabv 16899   normcnm 20167  NrmGrpcngp 20168  NrmRingcnrg 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-nrg 20176
This theorem is referenced by:  nrgabv  20240  nrgngp  20241  subrgnrg  20252  tngnrg  20253  cnnrg  20358  zhmnrg  26394
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