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Mirrors > Home > MPE Home > Th. List > crngmgp | Structured version Visualization version GIF version |
Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
ringmgp.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
crngmgp | ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringmgp.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
2 | 1 | iscrng 19233 | . 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) |
3 | 2 | simprbi 497 | 1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 CMndccmn 18835 mulGrpcmgp 19168 Ringcrg 19226 CRingccrg 19227 |
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 2790 |
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 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-cring 19229 |
This theorem is referenced by: crngcom 19241 gsummgp0 19287 prdscrngd 19292 crngbinom 19300 unitabl 19347 subrgcrng 19468 sraassa 20027 mplbas2 20179 evlslem3 20221 evlslem6 20222 evlslem1 20223 evlsgsummul 20233 evls1gsummul 20416 evl1gsummul 20451 mamuvs2 20943 matgsumcl 20997 madetsmelbas 21001 madetsmelbas2 21002 mdetleib2 21125 mdetf 21132 mdetdiaglem 21135 mdetdiag 21136 mdetdiagid 21137 mdetrlin 21139 mdetrsca 21140 mdetralt 21145 mdetuni0 21158 smadiadetlem4 21206 chpscmat 21378 chp0mat 21382 chpidmat 21383 amgmlem 25494 amgm 25495 wilthlem2 25573 wilthlem3 25574 lgseisenlem3 25880 lgseisenlem4 25881 cringm4 30880 mdetpmtr1 30987 mgpsumunsn 44337 mgpsumz 44338 mgpsumn 44339 amgmwlem 44831 amgmlemALT 44832 |
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