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| Mirrors > Home > MPE Home > Th. List > unitabl | Structured version Visualization version GIF version | ||
| Description: The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.) |
| Ref | Expression |
|---|---|
| unitmulcl.1 | β’ π = (Unitβπ ) |
| unitgrp.2 | β’ πΊ = ((mulGrpβπ ) βΎs π) |
| Ref | Expression |
|---|---|
| unitabl | β’ (π β CRing β πΊ β Abel) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | crngring 20147 | . . 3 β’ (π β CRing β π β Ring) | |
| 2 | unitmulcl.1 | . . . 4 β’ π = (Unitβπ ) | |
| 3 | unitgrp.2 | . . . 4 β’ πΊ = ((mulGrpβπ ) βΎs π) | |
| 4 | 2, 3 | unitgrp 20282 | . . 3 β’ (π β Ring β πΊ β Grp) |
| 5 | 1, 4 | syl 17 | . 2 β’ (π β CRing β πΊ β Grp) |
| 6 | eqid 2726 | . . . 4 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
| 7 | 6 | crngmgp 20143 | . . 3 β’ (π β CRing β (mulGrpβπ ) β CMnd) |
| 8 | 5 | grpmndd 18873 | . . 3 β’ (π β CRing β πΊ β Mnd) |
| 9 | 3 | subcmn 19754 | . . 3 β’ (((mulGrpβπ ) β CMnd β§ πΊ β Mnd) β πΊ β CMnd) |
| 10 | 7, 8, 9 | syl2anc 583 | . 2 β’ (π β CRing β πΊ β CMnd) |
| 11 | isabl 19701 | . 2 β’ (πΊ β Abel β (πΊ β Grp β§ πΊ β CMnd)) | |
| 12 | 5, 10, 11 | sylanbrc 582 | 1 β’ (π β CRing β πΊ β Abel) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 βΎs cress 17179 Mndcmnd 18664 Grpcgrp 18860 CMndccmn 19697 Abelcabl 19698 mulGrpcmgp 20036 Ringcrg 20135 CRingccrg 20136 Unitcui 20254 |
| 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-ress 17180 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-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 |
| This theorem is referenced by: cnmgpabl 21317 dchrpt 27150 |
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