drngnzr - Metamath Proof Explorer

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Theorem drngnzr 20604

Description: All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)

**Assertion**
Ref	Expression
drngnzr	⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)

**Proof of Theorem drngnzr**
Step	Hyp	Ref	Expression
1		drngring 20591	. 2 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		eqid 2726	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
3		eqid 2726	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
4	2, 3	drngunz 20603	. 2 ⊢ (𝑅 ∈ DivRing → (1_r‘𝑅) ≠ (0_g‘𝑅))
5	3, 2	isnzr 20413	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1_r‘𝑅) ≠ (0_g‘𝑅)))
6	1, 4, 5	sylanbrc 582	1 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2098 ≠ wne 2934 ‘cfv 6536 0_gc0g 17391 1_rcur 20083 Ringcrg 20135 NzRingcnzr 20411 DivRingcdr 20584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-nzr 20412 df-drng 20586

This theorem is referenced by: rng1nfld 20627 drngdomn 21212 islinds4 21725 drngidlhash 33057 drng0mxidl 33097 drngmxidl 33098 qsdrng 33116 ply1unit 33163 m1pmeq 33164 frlmdim 33213 ply1degltdimlem 33224 ply1degltdim 33225 fedgmul 33233 minplyirred 33289 algextdeglem4 33296 qqhnm 33499 lindsdom 36994 lindsenlbs 36995 matunitlindflem2 36997 0prjspnlem 41925 isldepslvec2 47423 lmod1zrnlvec 47432 aacllem 48104