Theorem rrextnrg 34336

Description: An extension of ℝ is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)

Assertion

Ref	Expression
rrextnrg	⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Proof of Theorem rrextnrg

Step	Hyp	Ref	Expression
1		eqid 2769	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2769	. . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3		eqid 2769	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 34335	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
5	4	simp1bi 1161	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
6	5	simpld 499	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   × cxp 5660   ↾ cres 5664  ‘cfv 6537  0cc0 11100  Basecbs 17269  distcds 17319  DivRingcdr 20813  metUnifcmetu 21482  ℤModczlm 21619  chrcchr 21620  UnifStcuss 24379  CUnifSpccusp 24422  NrmRingcnrg 24705  NrmModcnlm 24706   ℝExt crrext 34329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-res 5674  df-iota 6493  df-fv 6545  df-rrext 34334

This theorem is referenced by:  rrexttps  34341  rrexthaus  34342  rrhfe  34347  rrhcne  34348  rrhqima  34349  sitgclg  34677