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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 | ⊢ 𝑈 = (Unit‘𝑅) |
unitgrp.2 | ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) |
Ref | Expression |
---|---|
unitabl | ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 18558 | . . 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 18667 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Grp) |
6 | eqid 2622 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | 6 | crngmgp 18555 | . . 3 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd) |
8 | grpmnd 17429 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
9 | 5, 8 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd) |
10 | 3 | subcmn 18242 | . . 3 ⊢ (((mulGrp‘𝑅) ∈ CMnd ∧ 𝐺 ∈ Mnd) → 𝐺 ∈ CMnd) |
11 | 7, 9, 10 | syl2anc 693 | . 2 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
12 | isabl 18197 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
13 | 5, 11, 12 | sylanbrc 698 | 1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ↾s cress 15858 Mndcmnd 17294 Grpcgrp 17422 CMndccmn 18193 Abelcabl 18194 mulGrpcmgp 18489 Ringcrg 18547 CRingccrg 18548 Unitcui 18639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 |
This theorem is referenced by: cnmgpabl 19807 dchrpt 24992 |
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