Theorem rrextnlm 33986

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 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2729	. . . 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 33983	. . 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 1540   ∈ wcel 2109   × cxp 5629   ↾ cres 5633  ‘cfv 6499  0cc0 11044  Basecbs 17155  distcds 17205  DivRingcdr 20649  metUnifcmetu 21287  ℤModczlm 21442  chrcchr 21443  UnifStcuss 24174  CUnifSpccusp 24217  NrmRingcnrg 24500  NrmModcnlm 24501   ℝExt crrext 33977

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-res 5643  df-iota 6452  df-fv 6507  df-rrext 33982

This theorem is referenced by:  rrhfe  33995  rrhcne  33996  rrhqima  33997  sitgclg  34326