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Mirrors > Home > MPE Home > Th. List > idghm | Structured version Unicode version |
Description: The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
idghm.b |
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Ref | Expression |
---|---|
idghm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 |
. . 3
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2 | 1 | ancli 551 |
. 2
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3 | idghm.b |
. . . . . . . 8
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4 | eqid 2451 |
. . . . . . . 8
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5 | 3, 4 | grpcl 15662 |
. . . . . . 7
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6 | 5 | 3expb 1189 |
. . . . . 6
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7 | fvresi 6006 |
. . . . . 6
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8 | 6, 7 | syl 16 |
. . . . 5
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9 | fvresi 6006 |
. . . . . . 7
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10 | fvresi 6006 |
. . . . . . 7
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11 | 9, 10 | oveqan12d 6212 |
. . . . . 6
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12 | 11 | adantl 466 |
. . . . 5
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13 | 8, 12 | eqtr4d 2495 |
. . . 4
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14 | 13 | ralrimivva 2907 |
. . 3
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15 | f1oi 5777 |
. . . 4
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16 | f1of 5742 |
. . . 4
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17 | 15, 16 | ax-mp 5 |
. . 3
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18 | 14, 17 | jctil 537 |
. 2
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19 | 3, 3, 4, 4 | isghm 15858 |
. 2
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20 | 2, 18, 19 | sylanbrc 664 |
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-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 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-pw 3963 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-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-mnd 15526 df-grp 15656 df-ghm 15856 |
This theorem is referenced by: gicref 15910 symgga 16022 0frgp 16389 idlmhm 17237 frgpcyg 18124 nmoid 20446 idnghm 20447 |
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