![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > unitabl | GIF version |
Description: The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
unitgrp.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 13189 | . . 3 β’ (π β CRing β π β Ring) | |
2 | unitgrp.1 | . . . 4 β’ π = (Unitβπ ) | |
3 | unitgrp.2 | . . . 4 β’ πΊ = ((mulGrpβπ ) βΎs π) | |
4 | 2, 3 | unitgrp 13283 | . . 3 β’ (π β Ring β πΊ β Grp) |
5 | 1, 4 | syl 14 | . 2 β’ (π β CRing β πΊ β Grp) |
6 | 3 | a1i 9 | . . 3 β’ (π β CRing β πΊ = ((mulGrpβπ ) βΎs π)) |
7 | eqid 2177 | . . . 4 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
8 | 7 | crngmgp 13185 | . . 3 β’ (π β CRing β (mulGrpβπ ) β CMnd) |
9 | 5 | grpmndd 12888 | . . 3 β’ (π β CRing β πΊ β Mnd) |
10 | basfn 12519 | . . . . 5 β’ Base Fn V | |
11 | elex 2748 | . . . . 5 β’ (π β CRing β π β V) | |
12 | funfvex 5532 | . . . . . 6 β’ ((Fun Base β§ π β dom Base) β (Baseβπ ) β V) | |
13 | 12 | funfni 5316 | . . . . 5 β’ ((Base Fn V β§ π β V) β (Baseβπ ) β V) |
14 | 10, 11, 13 | sylancr 414 | . . . 4 β’ (π β CRing β (Baseβπ ) β V) |
15 | eqidd 2178 | . . . . 5 β’ (π β CRing β (Baseβπ ) = (Baseβπ )) | |
16 | 2 | a1i 9 | . . . . 5 β’ (π β CRing β π = (Unitβπ )) |
17 | ringsrg 13222 | . . . . . 6 β’ (π β Ring β π β SRing) | |
18 | 1, 17 | syl 14 | . . . . 5 β’ (π β CRing β π β SRing) |
19 | 15, 16, 18 | unitssd 13276 | . . . 4 β’ (π β CRing β π β (Baseβπ )) |
20 | 14, 19 | ssexd 4143 | . . 3 β’ (π β CRing β π β V) |
21 | 6, 8, 9, 20 | subcmnd 13127 | . 2 β’ (π β CRing β πΊ β CMnd) |
22 | isabl 13090 | . 2 β’ (πΊ β Abel β (πΊ β Grp β§ πΊ β CMnd)) | |
23 | 5, 21, 22 | sylanbrc 417 | 1 β’ (π β CRing β πΊ β Abel) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2737 Fn wfn 5211 βcfv 5216 (class class class)co 5874 Basecbs 12461 βΎs cress 12462 Grpcgrp 12876 CMndccmn 13086 Abelcabl 13087 mulGrpcmgp 13128 SRingcsrg 13144 Ringcrg 13177 CRingccrg 13178 Unitcui 13254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-tpos 6245 df-pnf 7993 df-mnf 7994 df-ltxr 7996 df-inn 8919 df-2 8977 df-3 8978 df-ndx 12464 df-slot 12465 df-base 12467 df-sets 12468 df-iress 12469 df-plusg 12548 df-mulr 12549 df-0g 12706 df-mgm 12774 df-sgrp 12807 df-mnd 12817 df-grp 12879 df-minusg 12880 df-cmn 13088 df-abl 13089 df-mgp 13129 df-ur 13141 df-srg 13145 df-ring 13179 df-cring 13180 df-oppr 13238 df-dvdsr 13256 df-unit 13257 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |