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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 20178 | . . 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 20315 | . . 3 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Grp) |
6 | eqid 2727 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
7 | 6 | crngmgp 20174 | . . 3 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd) |
8 | 5 | grpmndd 18896 | . . 3 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Mnd) |
9 | 3 | subcmn 19785 | . . 3 ⊢ (((mulGrp‘𝑅) ∈ CMnd ∧ 𝐺 ∈ Mnd) → 𝐺 ∈ CMnd) |
10 | 7, 8, 9 | syl2anc 583 | . 2 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) |
11 | isabl 19732 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
12 | 5, 10, 11 | sylanbrc 582 | 1 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 ↾s cress 17202 Mndcmnd 18687 Grpcgrp 18883 CMndccmn 19728 Abelcabl 19729 mulGrpcmgp 20067 Ringcrg 20166 CRingccrg 20167 Unitcui 20287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20266 df-dvdsr 20289 df-unit 20290 |
This theorem is referenced by: cnmgpabl 21354 dchrpt 27193 |
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