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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 2730 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 33278 | . . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷)))) |
5 | 4 | simp3bi 1145 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))) |
6 | 5 | simprd 494 | 1 ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 × cxp 5673 ↾ cres 5677 ‘cfv 6542 0cc0 11112 Basecbs 17148 distcds 17210 DivRingcdr 20500 metUnifcmetu 21135 ℤModczlm 21269 chrcchr 21270 UnifStcuss 23978 CUnifSpccusp 24022 NrmRingcnrg 24308 NrmModcnlm 24309 ℝExt crrext 33272 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-res 5687 df-iota 6494 df-fv 6550 df-rrext 33277 |
This theorem is referenced by: rrhfe 33290 rrhcne 33291 sitgclg 33639 |
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