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 2736 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 32248 | . . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))) |
5 | 4 | simp3bi 1146 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))) |
6 | 5 | simprd 496 | 1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 × cxp 5618 ↾ cres 5622 ‘cfv 6479 0cc0 10972 Basecbs 17009 distcds 17068 DivRingcdr 20093 metUnifcmetu 20694 ℤModczlm 20808 chrcchr 20809 UnifStcuss 23511 CUnifSpccusp 23555 NrmRingcnrg 23841 NrmModcnlm 23842 ℝExt crrext 32242 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-xp 5626 df-res 5632 df-iota 6431 df-fv 6487 df-rrext 32247 |
This theorem is referenced by: rrhfe 32260 rrhcne 32261 sitgclg 32609 |
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