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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrextust | Structured version Visualization version GIF version |
Description: The uniformity of an extension of ℝ is the uniformity generated by its distance. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
rrextust.b | ⊢ 𝐵 = (Base‘𝑅) |
rrextust.d | ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) |
Ref | Expression |
---|---|
rrextust | ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrextust.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | rrextust.d | . . . 4 ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵)) | |
3 | eqid 2733 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 32980 | . . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))) |
5 | 4 | simp3bi 1148 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))) |
6 | 5 | simprd 497 | 1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 × cxp 5675 ↾ cres 5679 ‘cfv 6544 0cc0 11110 Basecbs 17144 distcds 17206 DivRingcdr 20357 metUnifcmetu 20935 ℤModczlm 21050 chrcchr 21051 UnifStcuss 23758 CUnifSpccusp 23802 NrmRingcnrg 24088 NrmModcnlm 24089 ℝExt crrext 32974 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-res 5689 df-iota 6496 df-fv 6552 df-rrext 32979 |
This theorem is referenced by: rrhfe 32992 rrhcne 32993 sitgclg 33341 |
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