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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 20192 | . . 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 20329 | . . 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 2728 | . . . 4 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
| 7 | 6 | crngmgp 20188 | . . 3 β’ (π β CRing β (mulGrpβπ ) β CMnd) |
| 8 | 5 | grpmndd 18910 | . . 3 β’ (π β CRing β πΊ β Mnd) |
| 9 | 3 | subcmn 19799 | . . 3 β’ (((mulGrpβπ ) β CMnd β§ πΊ β Mnd) β πΊ β CMnd) |
| 10 | 7, 8, 9 | syl2anc 582 | . 2 β’ (π β CRing β πΊ β CMnd) |
| 11 | isabl 19746 | . 2 β’ (πΊ β Abel β (πΊ β Grp β§ πΊ β CMnd)) | |
| 12 | 5, 10, 11 | sylanbrc 581 | 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 6553 (class class class)co 7426 βΎs cress 17216 Mndcmnd 18701 Grpcgrp 18897 CMndccmn 19742 Abelcabl 19743 mulGrpcmgp 20081 Ringcrg 20180 CRingccrg 20181 Unitcui 20301 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 |
| This theorem is referenced by: cnmgpabl 21368 dchrpt 27220 |
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