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Theorem isnrg 18187
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 5541 . . . 4  |-  ( r  =  R  ->  ( norm `  r )  =  ( norm `  R
) )
2 isnrg.1 . . . 4  |-  N  =  ( norm `  R
)
31, 2syl6eqr 2346 . . 3  |-  ( r  =  R  ->  ( norm `  r )  =  N )
4 fveq2 5541 . . . 4  |-  ( r  =  R  ->  (AbsVal `  r )  =  (AbsVal `  R ) )
5 isnrg.2 . . . 4  |-  A  =  (AbsVal `  R )
64, 5syl6eqr 2346 . . 3  |-  ( r  =  R  ->  (AbsVal `  r )  =  A )
73, 6eleq12d 2364 . 2  |-  ( r  =  R  ->  (
( norm `  r )  e.  (AbsVal `  r )  <->  N  e.  A ) )
8 df-nrg 18124 . 2  |- NrmRing  =  {
r  e. NrmGrp  |  ( norm `  r )  e.  (AbsVal `  r ) }
97, 8elrab2 2938 1  |-  ( R  e. NrmRing 
<->  ( R  e. NrmGrp  /\  N  e.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  AbsValcabv 15597   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118
This theorem is referenced by:  nrgabv  18188  nrgngp  18189  subrgnrg  18200  tngnrg  18201  cnnrg  18306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-nrg 18124
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