Theorem rrextnlm 34199

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 2741	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2741	. . . 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 34196	. . 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 1153	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0))
6	5	simpld 496	1 ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   × cxp 5619   ↾ cres 5623  ‘cfv 6489  0cc0 11033  Basecbs 17174  distcds 17224  DivRingcdr 20705  metUnifcmetu 21342  ℤModczlm 21479  chrcchr 21480  UnifStcuss 24240  CUnifSpccusp 24283  NrmRingcnrg 24566  NrmModcnlm 24567   ℝExt crrext 34190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-xp 5627  df-res 5633  df-iota 6445  df-fv 6497  df-rrext 34195

This theorem is referenced by:  rrhfe  34208  rrhcne  34209  rrhqima  34210  sitgclg  34538