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Mirrors > Home > MPE Home > Th. List > grpo2grp | Structured version Unicode version |
Description: Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grp2grp.a |
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grp2grp.p |
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grp2grp.g |
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Ref | Expression |
---|---|
grpo2grp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grp2grp.a |
. . 3
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2 | 1 | eqcomi 2463 |
. 2
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3 | grp2grp.p |
. . 3
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4 | 3 | eqcomi 2463 |
. 2
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5 | grp2grp.g |
. . 3
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6 | eqid 2451 |
. . . 4
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7 | 6 | grpocl 23819 |
. . 3
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8 | 5, 7 | mp3an1 1302 |
. 2
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9 | 6 | grpoass 23822 |
. . 3
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10 | 5, 9 | mpan 670 |
. 2
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11 | eqid 2451 |
. . . 4
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12 | 6, 11 | grpoidcl 23836 |
. . 3
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13 | 5, 12 | ax-mp 5 |
. 2
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14 | 6, 11 | grpolid 23838 |
. . 3
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15 | 5, 14 | mpan 670 |
. 2
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16 | eqid 2451 |
. . . 4
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17 | 6, 16 | grpoinvcl 23845 |
. . 3
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18 | 5, 17 | mpan 670 |
. 2
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19 | 6, 11, 16 | grpolinv 23847 |
. . 3
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20 | 5, 19 | mpan 670 |
. 2
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21 | 2, 4, 8, 10, 13, 15, 18, 20 | isgrpi 15663 |
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 4498 ax-sep 4508 ax-nul 4516 ax-pow 4565 ax-pr 4626 ax-un 6469 |
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 2599 df-ne 2644 df-ral 2798 df-rex 2799 df-reu 2800 df-rmo 2801 df-rab 2802 df-v 3067 df-sbc 3282 df-csb 3384 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-nul 3733 df-if 3887 df-sn 3973 df-pr 3975 df-op 3979 df-uni 4187 df-iun 4268 df-br 4388 df-opab 4446 df-mpt 4447 df-id 4731 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fn 5516 df-f 5517 df-f1 5518 df-fo 5519 df-f1o 5520 df-fv 5521 df-riota 6148 df-ov 6190 df-0g 14479 df-mnd 15514 df-grp 15644 df-grpo 23810 df-gid 23811 df-ginv 23812 |
This theorem is referenced by: (None) |
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