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Theorem rrextnlm 31953
Description: The norm of an extension of 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
StepHypRef Expression
1 eqid 2738 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2738 . . . 4 ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3 rrextnlm.z . . . 4 𝑍 = (ℤMod‘𝑅)
41, 2, 3isrrext 31950 . . 3 (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
54simp2bi 1145 . 2 (𝑅 ∈ ℝExt → (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0))
65simpld 495 1 (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106   × cxp 5587  cres 5591  cfv 6433  0cc0 10871  Basecbs 16912  distcds 16971  DivRingcdr 19991  metUnifcmetu 20588  ℤModczlm 20702  chrcchr 20703  UnifStcuss 23405  CUnifSpccusp 23449  NrmRingcnrg 23735  NrmModcnlm 23736   ℝExt crrext 31944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601  df-iota 6391  df-fv 6441  df-rrext 31949
This theorem is referenced by:  rrhfe  31962  rrhcne  31963  rrhqima  31964  sitgclg  32309
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