Theorem rrextnlm 34162

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 2736	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2736	. . . 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 34159	. . 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 1146	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0))
6	5	simpld 494	1 ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   × cxp 5622   ↾ cres 5626  ‘cfv 6492  0cc0 11028  Basecbs 17138  distcds 17188  DivRingcdr 20664  metUnifcmetu 21302  ℤModczlm 21457  chrcchr 21458  UnifStcuss 24199  CUnifSpccusp 24242  NrmRingcnrg 24525  NrmModcnlm 24526   ℝExt crrext 34153

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-res 5636  df-iota 6448  df-fv 6500  df-rrext 34158

This theorem is referenced by:  rrhfe  34171  rrhcne  34172  rrhqima  34173  sitgclg  34501