Theorem rrextnlm 34302

Description: The norm of an extension of ℝ is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.)

Hypothesis

Ref	Expression
rrextnlm.z	⊢ 𝑍 = (ℤMod‘𝑅)

Assertion

Ref	Expression
rrextnlm	⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

Proof of Theorem rrextnlm

Step	Hyp	Ref	Expression
1		eqid 2764	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2764	. . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3		rrextnlm.z	. . . 4 ⊢ 𝑍 = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 34299	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
5	4	simp2bi 1160	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0))
6	5	simpld 498	1 ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144   × cxp 5647   ↾ cres 5651  ‘cfv 6523  0cc0 11075  Basecbs 17247  distcds 17297  DivRingcdr 20781  metUnifcmetu 21417  ℤModczlm 21554  chrcchr 21555  UnifStcuss 24315  CUnifSpccusp 24358  NrmRingcnrg 24641  NrmModcnlm 24642   ℝExt crrext 34293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-res 5661  df-iota 6479  df-fv 6531  df-rrext 34298

This theorem is referenced by:  rrhfe  34311  rrhcne  34312  rrhqima  34313  sitgclg  34641