Theorem rrextnrg 33974

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 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2729	. . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3		eqid 2729	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 33973	. . 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 1145	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
6	5	simpld 494	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   × cxp 5617   ↾ cres 5621  ‘cfv 6482  0cc0 11009  Basecbs 17120  distcds 17170  DivRingcdr 20614  metUnifcmetu 21252  ℤModczlm 21407  chrcchr 21408  UnifStcuss 24139  CUnifSpccusp 24182  NrmRingcnrg 24465  NrmModcnlm 24466   ℝExt crrext 33967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-res 5631  df-iota 6438  df-fv 6490  df-rrext 33972

This theorem is referenced by:  rrexttps  33979  rrexthaus  33980  rrhfe  33985  rrhcne  33986  rrhqima  33987  sitgclg  34316