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Mirrors > Home > MPE Home > Th. List > grpinveu | Structured version Visualization version Unicode version |
Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.) |
Ref | Expression |
---|---|
grpinveu.b |
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grpinveu.p |
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grpinveu.o |
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Ref | Expression |
---|---|
grpinveu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinveu.b |
. . . 4
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2 | grpinveu.p |
. . . 4
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3 | grpinveu.o |
. . . 4
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4 | 1, 2, 3 | grpinvex 16759 |
. . 3
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5 | eqtr3 2492 |
. . . . . . . . . . . 12
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6 | 1, 2 | grprcan 16777 |
. . . . . . . . . . . 12
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7 | 5, 6 | syl5ib 227 |
. . . . . . . . . . 11
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8 | 7 | 3exp2 1251 |
. . . . . . . . . 10
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9 | 8 | com24 89 |
. . . . . . . . 9
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10 | 9 | imp41 604 |
. . . . . . . 8
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11 | 10 | an32s 821 |
. . . . . . 7
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12 | 11 | expd 443 |
. . . . . 6
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13 | 12 | ralrimdva 2812 |
. . . . 5
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14 | 13 | ancld 562 |
. . . 4
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15 | 14 | reximdva 2858 |
. . 3
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16 | 4, 15 | mpd 15 |
. 2
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17 | oveq1 6315 |
. . . 4
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18 | 17 | eqeq1d 2473 |
. . 3
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19 | 18 | reu8 3222 |
. 2
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20 | 16, 19 | sylibr 217 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-iota 5553 df-fun 5591 df-fv 5597 df-riota 6270 df-ov 6311 df-0g 15418 df-mgm 16566 df-sgrp 16605 df-mnd 16615 df-grp 16751 |
This theorem is referenced by: grpinvf 16788 grplinv 16790 isgrpinv 16794 |
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