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 2824 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2824 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2824 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31245 | . . 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 1141 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing)) |
6 | 5 | simpld 497 | 1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 × cxp 5556 ↾ cres 5560 ‘cfv 6358 0cc0 10540 Basecbs 16486 distcds 16577 DivRingcdr 19505 metUnifcmetu 20539 ℤModczlm 20651 chrcchr 20652 UnifStcuss 22865 CUnifSpccusp 22909 NrmRingcnrg 23192 NrmModcnlm 23193 ℝExt crrext 31239 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-xp 5564 df-res 5570 df-iota 6317 df-fv 6366 df-rrext 31244 |
This theorem is referenced by: rrexttps 31251 rrexthaus 31252 rrhfe 31257 rrhcne 31258 rrhqima 31259 sitgclg 31604 |
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