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Mirrors > Home > ILE Home > Th. List > grprinvd | Unicode version |
Description: Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grprinvlem.c |
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grprinvlem.o |
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grprinvlem.i |
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grprinvlem.a |
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grprinvlem.n |
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grprinvd.x |
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grprinvd.n |
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grprinvd.e |
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Ref | Expression |
---|---|
grprinvd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grprinvlem.c |
. 2
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2 | grprinvlem.o |
. 2
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3 | grprinvlem.i |
. 2
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4 | grprinvlem.a |
. 2
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5 | grprinvlem.n |
. 2
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6 | 1 | 3expb 1204 |
. . . . 5
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7 | 6 | caovclg 6026 |
. . . 4
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8 | 7 | adantlr 477 |
. . 3
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9 | grprinvd.x |
. . 3
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10 | grprinvd.n |
. . 3
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11 | 8, 9, 10 | caovcld 6027 |
. 2
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12 | 4 | caovassg 6032 |
. . . . 5
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13 | 12 | adantlr 477 |
. . . 4
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14 | 13, 9, 10, 11 | caovassd 6033 |
. . 3
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15 | grprinvd.e |
. . . . . 6
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16 | 15 | oveq1d 5889 |
. . . . 5
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17 | 13, 10, 9, 10 | caovassd 6033 |
. . . . 5
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18 | oveq2 5882 |
. . . . . . 7
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19 | id 19 |
. . . . . . 7
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20 | 18, 19 | eqeq12d 2192 |
. . . . . 6
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21 | 3 | ralrimiva 2550 |
. . . . . . . 8
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22 | oveq2 5882 |
. . . . . . . . . 10
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23 | id 19 |
. . . . . . . . . 10
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24 | 22, 23 | eqeq12d 2192 |
. . . . . . . . 9
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25 | 24 | cbvralvw 2707 |
. . . . . . . 8
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26 | 21, 25 | sylib 122 |
. . . . . . 7
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27 | 26 | adantr 276 |
. . . . . 6
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28 | 20, 27, 10 | rspcdva 2846 |
. . . . 5
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29 | 16, 17, 28 | 3eqtr3d 2218 |
. . . 4
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30 | 29 | oveq2d 5890 |
. . 3
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31 | 14, 30 | eqtrd 2210 |
. 2
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32 | 1, 2, 3, 4, 5, 11, 31 | grprinvlem 12803 |
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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 |
This theorem is referenced by: grpridd 12805 grprcan 12909 grprinv 12922 |
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