Theorem rrextnrg 31242

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 2821	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2821	. . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3		eqid 2821	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 31241	. . 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 1141	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
6	5	simpld 497	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   × cxp 5553   ↾ cres 5557  ‘cfv 6355  0cc0 10537  Basecbs 16483  distcds 16574  DivRingcdr 19502  metUnifcmetu 20536  ℤModczlm 20648  chrcchr 20649  UnifStcuss 22862  CUnifSpccusp 22906  NrmRingcnrg 23189  NrmModcnlm 23190   ℝExt crrext 31235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-res 5567  df-iota 6314  df-fv 6363  df-rrext 31240

This theorem is referenced by:  rrexttps  31247  rrexthaus  31248  rrhfe  31253  rrhcne  31254  rrhqima  31255  sitgclg  31600