Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrextcusp | Structured version Visualization version GIF version |
Description: An extension of ℝ is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
rrextcusp | ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2818 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2818 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31140 | . . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))) |
5 | 4 | simp3bi 1139 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))) |
6 | 5 | simpld 495 | 1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 × cxp 5546 ↾ cres 5550 ‘cfv 6348 0cc0 10525 Basecbs 16471 distcds 16562 DivRingcdr 19431 metUnifcmetu 20464 ℤModczlm 20576 chrcchr 20577 UnifStcuss 22789 CUnifSpccusp 22833 NrmRingcnrg 23116 NrmModcnlm 23117 ℝExt crrext 31134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-res 5560 df-iota 6307 df-fv 6356 df-rrext 31139 |
This theorem is referenced by: rrhfe 31152 rrhcne 31153 sitgclg 31499 |
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