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Mirrors > Home > ILE Home > Th. List > eqgid | Unicode version |
Description: The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.) |
Ref | Expression |
---|---|
eqger.x |
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eqger.r |
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eqgid.3 |
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Ref | Expression |
---|---|
eqgid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 12970 |
. . . . 5
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2 | eqger.r |
. . . . . 6
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3 | 2 | releqgg 13011 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | mpancom 422 |
. . . 4
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5 | relelec 6572 |
. . . 4
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6 | 4, 5 | syl 14 |
. . 3
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7 | 1 | adantr 276 |
. . . . . . . . 9
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8 | eqgid.3 |
. . . . . . . . . 10
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9 | eqid 2177 |
. . . . . . . . . 10
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10 | 8, 9 | grpinvid 12862 |
. . . . . . . . 9
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11 | 7, 10 | syl 14 |
. . . . . . . 8
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12 | 11 | oveq1d 5887 |
. . . . . . 7
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13 | eqger.x |
. . . . . . . . 9
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14 | eqid 2177 |
. . . . . . . . 9
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15 | 13, 14, 8 | grplid 12838 |
. . . . . . . 8
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16 | 1, 15 | sylan 283 |
. . . . . . 7
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17 | 12, 16 | eqtrd 2210 |
. . . . . 6
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18 | 17 | eleq1d 2246 |
. . . . 5
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19 | 18 | pm5.32da 452 |
. . . 4
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20 | 13 | subgss 12965 |
. . . . 5
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21 | 13, 8 | grpidcl 12836 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 1, 21 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 13, 9, 14, 2 | eqgval 13013 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 3anass 982 |
. . . . . . 7
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25 | 23, 24 | bitrdi 196 |
. . . . . 6
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26 | 25 | baibd 923 |
. . . . 5
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27 | 1, 20, 22, 26 | syl21anc 1237 |
. . . 4
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28 | 20 | sseld 3154 |
. . . . 5
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29 | 28 | pm4.71rd 394 |
. . . 4
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30 | 19, 27, 29 | 3bitr4d 220 |
. . 3
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31 | 6, 30 | bitrd 188 |
. 2
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32 | 31 | eqrdv 2175 |
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-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
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-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-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-ec 6534 df-inn 8916 df-2 8974 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-subg 12961 df-eqg 12963 |
This theorem is referenced by: (None) |
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