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Mirrors > Home > MPE Home > Th. List > grpinvid | Structured version Unicode version |
Description: The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.) |
Ref | Expression |
---|---|
grpinvid.u |
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grpinvid.n |
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Ref | Expression |
---|---|
grpinvid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2454 |
. . . 4
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2 | grpinvid.u |
. . . 4
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3 | 1, 2 | grpidcl 15686 |
. . 3
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4 | eqid 2454 |
. . . 4
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5 | 1, 4, 2 | grplid 15688 |
. . 3
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6 | 3, 5 | mpdan 668 |
. 2
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7 | grpinvid.n |
. . . 4
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8 | 1, 4, 2, 7 | grpinvid1 15706 |
. . 3
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9 | 3, 3, 8 | mpd3an23 1317 |
. 2
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10 | 6, 9 | mpbird 232 |
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 1955 ax-ext 2432 ax-rep 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 |
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 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-sn 3987 df-pr 3989 df-op 3993 df-uni 4201 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-id 4745 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-riota 6162 df-ov 6204 df-0g 14500 df-mnd 15535 df-grp 15665 df-minusg 15666 |
This theorem is referenced by: grpinvnz 15717 grpsubid1 15731 mulgneg 15765 mulgz 15768 0subg 15826 eqgid 15853 odnncl 16170 gexdvds 16205 gsumzinv 16565 gsumzinvOLD 16566 gsumsub 16570 gsumsubOLD 16571 dprdfinv 16632 dprdfinvOLD 16639 mplsubglem 17635 mplsubglemOLD 17637 dsmmsubg 18294 dchrisum0re 22896 baerlem3lem1 35691 |
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