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Mirrors > Home > ILE Home > Th. List > grpinvfng | Unicode version |
Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
grpinvfn.b |
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grpinvfn.n |
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Ref | Expression |
---|---|
grpinvfng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvfn.b |
. . . . . 6
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2 | basfn 12512 |
. . . . . . 7
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3 | elex 2748 |
. . . . . . 7
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4 | funfvex 5531 |
. . . . . . . 8
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5 | 4 | funfni 5315 |
. . . . . . 7
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6 | 2, 3, 5 | sylancr 414 |
. . . . . 6
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7 | 1, 6 | eqeltrid 2264 |
. . . . 5
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8 | riotaexg 5832 |
. . . . 5
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9 | 7, 8 | syl 14 |
. . . 4
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10 | 9 | ralrimivw 2551 |
. . 3
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11 | eqid 2177 |
. . . 4
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12 | 11 | fnmpt 5341 |
. . 3
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13 | 10, 12 | syl 14 |
. 2
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14 | eqid 2177 |
. . . 4
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15 | eqid 2177 |
. . . 4
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16 | grpinvfn.n |
. . . 4
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17 | 1, 14, 15, 16 | grpinvfvalg 12847 |
. . 3
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18 | 17 | fneq1d 5305 |
. 2
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19 | 13, 18 | mpbird 167 |
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-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-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-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 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-inn 8916 df-ndx 12457 df-slot 12458 df-base 12460 df-minusg 12813 |
This theorem is referenced by: isgrpinv 12858 mulgval 12918 mulgfng 12919 invrfvald 13222 |
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