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Mirrors > Home > MPE Home > Th. List > drngring | Structured version Visualization version GIF version |
Description: A division ring is a ring. (Contributed by NM, 8-Sep-2011.) |
Ref | Expression |
---|---|
drngring | ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdrng 19437 | . 2 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = ((Base‘𝑅) ∖ {(0g‘𝑅)}))) |
5 | 4 | simplbi 498 | 1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∖ cdif 3932 {csn 4559 ‘cfv 6349 Basecbs 16473 0gc0g 16703 Ringcrg 19228 Unitcui 19320 DivRingcdr 19433 |
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 2793 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-iota 6308 df-fv 6357 df-drng 19435 |
This theorem is referenced by: drnggrp 19441 drngid 19447 drngunz 19448 drnginvrcl 19450 drnginvrn0 19451 drnginvrl 19452 drnginvrr 19453 drngmul0or 19454 sdrgid 19506 sdrgacs 19511 cntzsdrg 19512 primefld 19515 abvtriv 19543 rlmlvec 19909 drngnidl 19932 drnglpir 19956 drngnzr 19965 drngdomn 20006 mpllvec 20163 qsssubdrg 20534 frlmlvec 20835 frlmphllem 20854 frlmphl 20855 cvsdivcl 23666 qcvs 23680 cphsubrglem 23710 rrxcph 23924 rrx0 23929 drnguc1p 24693 ig1peu 24694 ig1pcl 24698 ig1pdvds 24699 ig1prsp 24700 ply1lpir 24701 padicabv 26134 ofldchr 30815 reofld 30841 rearchi 30843 xrge0slmod 30845 sradrng 30888 drgext0gsca 30894 drgextlsp 30896 rgmoddim 30908 frlmdim 30909 matdim 30913 drngdimgt0 30916 fedgmullem1 30925 fedgmullem2 30926 fedgmul 30927 fldextid 30949 extdg1id 30953 ccfldsrarelvec 30956 zrhunitpreima 31119 elzrhunit 31120 qqhval2lem 31122 qqh0 31125 qqh1 31126 qqhf 31127 qqhghm 31129 qqhrhm 31130 qqhnm 31131 qqhucn 31133 zrhre 31160 qqhre 31161 lindsdom 34768 lindsenlbs 34769 matunitlindflem1 34770 matunitlindflem2 34771 matunitlindf 34772 dvalveclem 38043 dvhlveclem 38126 hlhilsrnglem 38971 0prjspnrel 39149 drhmsubc 44249 drngcat 44250 drhmsubcALTV 44267 drngcatALTV 44268 aacllem 44800 |
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