![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | ⊢ 𝑈 = (Unit‘𝑅) |
unitgrp.2 | ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) |
Ref | Expression |
---|---|
unitabl | ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19044 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | unitmulcl.1 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | unitgrp.2 | . . . 4 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
4 | 2, 3 | unitgrp 19153 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Grp) |
6 | eqid 2773 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | 6 | crngmgp 19041 | . . 3 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd) |
8 | grpmnd 17911 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
9 | 5, 8 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd) |
10 | 3 | subcmn 18728 | . . 3 ⊢ (((mulGrp‘𝑅) ∈ CMnd ∧ 𝐺 ∈ Mnd) → 𝐺 ∈ CMnd) |
11 | 7, 9, 10 | syl2anc 576 | . 2 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
12 | isabl 18683 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
13 | 5, 11, 12 | sylanbrc 575 | 1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ‘cfv 6186 (class class class)co 6975 ↾s cress 16339 Mndcmnd 17775 Grpcgrp 17904 CMndccmn 18679 Abelcabl 18680 mulGrpcmgp 18975 Ringcrg 19033 CRingccrg 19034 Unitcui 19125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-tpos 7694 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-0g 16570 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-grp 17907 df-cmn 18681 df-abl 18682 df-mgp 18976 df-ur 18988 df-ring 19035 df-cring 19036 df-oppr 19109 df-dvdsr 19127 df-unit 19128 |
This theorem is referenced by: cnmgpabl 20324 dchrpt 25561 |
Copyright terms: Public domain | W3C validator |