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Mirrors > Home > ILE Home > Th. List > unitabl | Unicode version |
Description: The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
unitgrp.1 |
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unitgrp.2 |
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Ref | Expression |
---|---|
unitabl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 13122 |
. . 3
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2 | unitgrp.1 |
. . . 4
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3 | unitgrp.2 |
. . . 4
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4 | 2, 3 | unitgrp 13216 |
. . 3
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5 | 1, 4 | syl 14 |
. 2
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6 | 3 | a1i 9 |
. . 3
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7 | eqid 2177 |
. . . 4
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8 | 7 | crngmgp 13118 |
. . 3
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9 | 5 | grpmndd 12821 |
. . 3
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10 | basfn 12512 |
. . . . 5
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11 | elex 2748 |
. . . . 5
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12 | funfvex 5531 |
. . . . . 6
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13 | 12 | funfni 5315 |
. . . . 5
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14 | 10, 11, 13 | sylancr 414 |
. . . 4
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15 | eqidd 2178 |
. . . . 5
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16 | 2 | a1i 9 |
. . . . 5
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17 | ringsrg 13155 |
. . . . . 6
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18 | 1, 17 | syl 14 |
. . . . 5
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19 | 15, 16, 18 | unitssd 13209 |
. . . 4
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20 | 14, 19 | ssexd 4142 |
. . 3
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21 | 6, 8, 9, 20 | subcmnd 13060 |
. 2
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22 | isabl 13023 |
. 2
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23 | 5, 21, 22 | sylanbrc 417 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
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 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-tpos 6243 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-mulr 12542 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-cmn 13021 df-abl 13022 df-mgp 13062 df-ur 13074 df-srg 13078 df-ring 13112 df-cring 13113 df-oppr 13171 df-dvdsr 13189 df-unit 13190 |
This theorem is referenced by: (None) |
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