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Mirrors > Home > ILE Home > Th. List > grpinvadd | Unicode version |
Description: The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) |
Ref | Expression |
---|---|
grpinvadd.b |
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grpinvadd.p |
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grpinvadd.n |
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Ref | Expression |
---|---|
grpinvadd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 |
. . . 4
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2 | simp2 998 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | simp3 999 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | grpinvadd.b |
. . . . . . 7
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5 | grpinvadd.n |
. . . . . . 7
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6 | 4, 5 | grpinvcl 12808 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 6 | 3adant2 1016 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 4, 5 | grpinvcl 12808 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8 | 3adant3 1017 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | grpinvadd.p |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 4, 10 | grpcl 12772 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 1, 7, 9, 11 | syl3anc 1238 |
. . . 4
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13 | 4, 10 | grpass 12773 |
. . . 4
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14 | 1, 2, 3, 12, 13 | syl13anc 1240 |
. . 3
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15 | eqid 2177 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 4, 10, 15, 5 | grprinv 12810 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | 3adant2 1016 |
. . . . . 6
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18 | 17 | oveq1d 5884 |
. . . . 5
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19 | 4, 10 | grpass 12773 |
. . . . . 6
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20 | 1, 3, 7, 9, 19 | syl13anc 1240 |
. . . . 5
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21 | 4, 10, 15 | grplid 12793 |
. . . . . 6
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22 | 1, 9, 21 | syl2anc 411 |
. . . . 5
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23 | 18, 20, 22 | 3eqtr3d 2218 |
. . . 4
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24 | 23 | oveq2d 5885 |
. . 3
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25 | 4, 10, 15, 5 | grprinv 12810 |
. . . 4
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26 | 25 | 3adant3 1017 |
. . 3
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27 | 14, 24, 26 | 3eqtrd 2214 |
. 2
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28 | 4, 10 | grpcl 12772 |
. . 3
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29 | 4, 10, 15, 5 | grpinvid1 12811 |
. . 3
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30 | 1, 28, 12, 29 | syl3anc 1238 |
. 2
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31 | 27, 30 | mpbird 167 |
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-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 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-inn 8906 df-2 8964 df-ndx 12445 df-slot 12446 df-base 12448 df-plusg 12528 df-0g 12652 df-mgm 12664 df-sgrp 12697 df-mnd 12707 df-grp 12767 df-minusg 12768 |
This theorem is referenced by: grpinvsub 12838 mulgaddcomlem 12891 mulginvcom 12893 mulgdir 12900 ablinvadd 12937 ablsub2inv 12938 |
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