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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 18377 | . 2 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) |
3 | 2 | simprbi 479 | 1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 CMndccmn 18016 mulGrpcmgp 18312 Ringcrg 18370 CRingccrg 18371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-cring 18373 |
This theorem is referenced by: crngcom 18385 gsummgp0 18431 prdscrngd 18436 crngbinom 18444 unitabl 18491 subrgcrng 18607 sraassa 19146 mplbas2 19291 evlslem6 19334 evlslem3 19335 evlslem1 19336 evls1gsummul 19511 evl1gsummul 19545 mamuvs2 20031 matgsumcl 20085 madetsmelbas 20089 madetsmelbas2 20090 mdetleib2 20213 mdetf 20220 mdetdiaglem 20223 mdetdiag 20224 mdetdiagid 20225 mdetrlin 20227 mdetrsca 20228 mdetralt 20233 mdetuni0 20246 smadiadetlem4 20294 chpscmat 20466 chp0mat 20470 chpidmat 20471 amgmlem 24516 amgm 24517 wilthlem2 24595 wilthlem3 24596 lgseisenlem3 24902 lgseisenlem4 24903 mdetpmtr1 29217 mgpsumunsn 41933 mgpsumz 41934 mgpsumn 41935 amgmwlem 42357 amgmlemALT 42358 |
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