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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrextnrg | Structured version Visualization version GIF version |
Description: An extension of ℝ is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
rrextnrg | ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2798 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2798 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31351 | . . 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 | simp1bi 1142 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing)) |
6 | 5 | simpld 498 | 1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 × cxp 5517 ↾ cres 5521 ‘cfv 6324 0cc0 10526 Basecbs 16475 distcds 16566 DivRingcdr 19495 metUnifcmetu 20082 ℤModczlm 20194 chrcchr 20195 UnifStcuss 22859 CUnifSpccusp 22903 NrmRingcnrg 23186 NrmModcnlm 23187 ℝExt crrext 31345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-res 5531 df-iota 6283 df-fv 6332 df-rrext 31350 |
This theorem is referenced by: rrexttps 31357 rrexthaus 31358 rrhfe 31363 rrhcne 31364 rrhqima 31365 sitgclg 31710 |
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