Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrextdrg | Structured version Visualization version GIF version |
Description: An extension of ℝ is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
rrextdrg | ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2737 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2737 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31892 | . . 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 1143 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing)) |
6 | 5 | simprd 495 | 1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 × cxp 5583 ↾ cres 5587 ‘cfv 6423 0cc0 10818 Basecbs 16856 distcds 16915 DivRingcdr 19935 metUnifcmetu 20532 ℤModczlm 20646 chrcchr 20647 UnifStcuss 23349 CUnifSpccusp 23393 NrmRingcnrg 23679 NrmModcnlm 23680 ℝExt crrext 31886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3071 df-v 3429 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-xp 5591 df-res 5597 df-iota 6381 df-fv 6431 df-rrext 31891 |
This theorem is referenced by: rrhfe 31904 rrhcne 31905 rrhqima 31906 rrh0 31907 sitgclg 32251 |
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