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Mirrors > Home > MPE Home > Th. List > isgim | Structured version Unicode version |
Description: An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
isgim.b |
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isgim.c |
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Ref | Expression |
---|---|
isgim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 967 |
. 2
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2 | df-gim 15898 |
. . 3
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3 | ovex 6218 |
. . . 4
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4 | 3 | rabex 4544 |
. . 3
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5 | oveq12 6202 |
. . . 4
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6 | fveq2 5792 |
. . . . . 6
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7 | isgim.b |
. . . . . 6
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8 | 6, 7 | syl6eqr 2510 |
. . . . 5
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9 | fveq2 5792 |
. . . . . 6
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10 | isgim.c |
. . . . . 6
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11 | 9, 10 | syl6eqr 2510 |
. . . . 5
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12 | f1oeq23 5736 |
. . . . 5
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13 | 8, 11, 12 | syl2an 477 |
. . . 4
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14 | 5, 13 | rabeqbidv 3066 |
. . 3
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15 | 2, 4, 14 | elovmpt2 6410 |
. 2
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16 | ghmgrp1 15860 |
. . . . . 6
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17 | ghmgrp2 15861 |
. . . . . 6
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18 | 16, 17 | jca 532 |
. . . . 5
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19 | 18 | adantr 465 |
. . . 4
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20 | 19 | pm4.71ri 633 |
. . 3
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21 | f1oeq1 5733 |
. . . . 5
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22 | 21 | elrab 3217 |
. . . 4
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23 | 22 | anbi2i 694 |
. . 3
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24 | 20, 23 | bitr4i 252 |
. 2
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25 | 1, 15, 24 | 3bitr4i 277 |
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-ghm 15856 df-gim 15898 |
This theorem is referenced by: gimf1o 15902 gimghm 15903 isgim2 15904 invoppggim 15986 lmimgim 17261 zzngim 18103 cygznlem3 18120 reefgim 22041 imasgim 29596 pmattomply1grpiso 31274 |
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