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Theorem rrextnrg 27604
Description: An extension of  RR is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
Assertion
Ref Expression
rrextnrg  |-  ( R  e. ℝExt  ->  R  e. NrmRing )

Proof of Theorem rrextnrg
StepHypRef Expression
1 eqid 2460 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2460 . . . 4  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
3 eqid 2460 . . . 4  |-  ( ZMod
`  R )  =  ( ZMod `  R
)
41, 2, 3isrrext 27603 . . 3  |-  ( R  e. ℝExt 
<->  ( ( R  e. NrmRing  /\  R  e.  DivRing )  /\  ( ( ZMod `  R )  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( R  e. CUnifSp  /\  (UnifSt `  R
)  =  (metUnif `  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) ) ) )
54simp1bi 1006 . 2  |-  ( R  e. ℝExt  ->  ( R  e. NrmRing  /\  R  e.  DivRing ) )
65simpld 459 1  |-  ( R  e. ℝExt  ->  R  e. NrmRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    X. cxp 4990    |` cres 4994   ` cfv 5579   0cc0 9481   Basecbs 14479   distcds 14553   DivRingcdr 17172  metUnifcmetu 18174   ZModczlm 18298  chrcchr 18299  UnifStcuss 20484  CUnifSpccusp 20528  NrmRingcnrg 20828  NrmModcnlm 20829   ℝExt crrext 27597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-res 5004  df-iota 5542  df-fv 5587  df-rrext 27602
This theorem is referenced by:  rrexttps  27609  rrexthaus  27610  rrhfe  27614  rrhcne  27615  sitgclg  27910
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