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Mirrors > Home > MPE Home > Th. List > drngnzr | Structured version Visualization version GIF version |
Description: A division ring is a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
drngnzr | ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 20707 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2726 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | eqid 2726 | . . 3 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
4 | 2, 3 | drngunz 20718 | . 2 ⊢ (𝑅 ∈ DivRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
5 | 3, 2 | isnzr 20489 | . 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
6 | 1, 4, 5 | sylanbrc 581 | 1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6543 0gc0g 17446 1rcur 20157 Ringcrg 20209 NzRingcnzr 20487 DivRingcdr 20700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-plusg 17271 df-mulr 17272 df-0g 17448 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-grp 18923 df-minusg 18924 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20309 df-dvdsr 20332 df-unit 20333 df-nzr 20488 df-drng 20702 |
This theorem is referenced by: drngdomn 20720 rng1nfld 20751 islinds4 21826 drngidlhash 33312 drng0mxidl 33354 drngmxidl 33355 qsdrng 33375 ply1unit 33450 m1pmeq 33458 frlmdim 33509 ply1degltdimlem 33520 ply1degltdim 33521 fedgmul 33529 minplyirred 33583 algextdeglem4 33590 rtelextdg2lem 33596 2sqr3minply 33617 qqhnm 33815 lindsdom 37325 lindsenlbs 37326 matunitlindflem2 37328 aks6d1c2lem4 41836 aks6d1c5lem3 41846 aks6d1c6lem1 41879 0prjspnlem 42310 isldepslvec2 47901 lmod1zrnlvec 47910 aacllem 48582 |
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