Theorem rrextnlm 34417

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 2766	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2766	. . . 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 34414	. . 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 1164	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0))
6	5	simpld 500	1 ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   × cxp 5664   ↾ cres 5668  ‘cfv 6543  0cc0 11118  Basecbs 17294  distcds 17344  DivRingcdr 20857  metUnifcmetu 21543  ℤModczlm 21680  chrcchr 21681  UnifStcuss 24440  CUnifSpccusp 24483  NrmRingcnrg 24766  NrmModcnlm 24767   ℝExt crrext 34408

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-res 5678  df-iota 6499  df-fv 6551  df-rrext 34413

This theorem is used by:  rrhfe  34426  rrhcne  34427  rrhqima  34428  sitgclg  34756