Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > crngcom | Structured version Visualization version GIF version |
Description: A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
ringcl.b | โข ๐ต = (Baseโ๐ ) |
ringcl.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
crngcom | โข ((๐ โ CRing โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | 1 | crngmgp 19886 | . 2 โข (๐ โ CRing โ (mulGrpโ๐ ) โ CMnd) |
3 | ringcl.b | . . . 4 โข ๐ต = (Baseโ๐ ) | |
4 | 1, 3 | mgpbas 19821 | . . 3 โข ๐ต = (Baseโ(mulGrpโ๐ )) |
5 | ringcl.t | . . . 4 โข ยท = (.rโ๐ ) | |
6 | 1, 5 | mgpplusg 19819 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
7 | 4, 6 | cmncom 19498 | . 2 โข (((mulGrpโ๐ ) โ CMnd โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
8 | 2, 7 | syl3an1 1162 | 1 โข ((๐ โ CRing โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) = (๐ ยท ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1086 = wceq 1540 โ wcel 2105 โcfv 6479 (class class class)co 7337 Basecbs 17009 .rcmulr 17060 CMndccmn 19481 mulGrpcmgp 19815 CRingccrg 19879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-plusg 17072 df-cmn 19483 df-mgp 19816 df-cring 19881 |
This theorem is referenced by: crngoppr 19961 unitmulclb 20002 idsrngd 20228 rmodislmodlem 20296 lmhmvsca 20413 quscrng 20617 frlmphl 21094 psrcom 21284 psrass23 21285 mplcoe2 21348 mamutpos 21713 mat1dimcrng 21732 dmatcrng 21757 scmatcrng 21776 mdetrsca 21858 mdetmul 21878 cramerimp 21941 fta1glem1 25436 rdivmuldivd 31775 ringlsmss1 31881 mxidlprm 31937 extdg1id 32036 mdetlap 32080 rncrhmcl 40500 mhphf 40545 |
Copyright terms: Public domain | W3C validator |