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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 2736 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2736 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2736 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 32450 | . . 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 1145 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing)) |
6 | 5 | simpld 495 | 1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 × cxp 5629 ↾ cres 5633 ‘cfv 6493 0cc0 11047 Basecbs 17075 distcds 17134 DivRingcdr 20170 metUnifcmetu 20772 ℤModczlm 20886 chrcchr 20887 UnifStcuss 23589 CUnifSpccusp 23633 NrmRingcnrg 23919 NrmModcnlm 23920 ℝExt crrext 32444 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-res 5643 df-iota 6445 df-fv 6501 df-rrext 32449 |
This theorem is referenced by: rrexttps 32456 rrexthaus 32457 rrhfe 32462 rrhcne 32463 rrhqima 32464 sitgclg 32811 |
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