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Mirrors > Home > MPE Home > Th. List > grpoinveu | Structured 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, 27-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinveu.1 |
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grpinveu.2 |
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Ref | Expression |
---|---|
grpoinveu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinveu.1 |
. . . . 5
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2 | grpinveu.2 |
. . . . 5
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3 | 1, 2 | grpoidinv2 23850 |
. . . 4
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4 | simpl 457 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | reximi 2922 |
. . . . 5
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6 | 5 | adantl 466 |
. . . 4
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7 | 3, 6 | syl 16 |
. . 3
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8 | eqtr3 2479 |
. . . . . . . . . . . 12
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9 | 1 | grporcan 23853 |
. . . . . . . . . . . 12
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10 | 8, 9 | syl5ib 219 |
. . . . . . . . . . 11
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11 | 10 | 3exp2 1206 |
. . . . . . . . . 10
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12 | 11 | com24 87 |
. . . . . . . . 9
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13 | 12 | imp41 593 |
. . . . . . . 8
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14 | 13 | an32s 802 |
. . . . . . 7
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15 | 14 | expd 436 |
. . . . . 6
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16 | 15 | ralrimdva 2905 |
. . . . 5
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17 | 16 | ancld 553 |
. . . 4
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18 | 17 | reximdva 2927 |
. . 3
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19 | 7, 18 | mpd 15 |
. 2
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20 | oveq1 6200 |
. . . 4
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21 | 20 | eqeq1d 2453 |
. . 3
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22 | 21 | reu8 3255 |
. 2
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23 | 19, 22 | sylibr 212 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pr 4632 ax-un 6475 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-fo 5525 df-fv 5527 df-riota 6154 df-ov 6196 df-grpo 23823 df-gid 23824 |
This theorem is referenced by: grpoinvcl 23858 grpoinv 23859 |
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