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Mirrors > Home > ILE Home > Th. List > grpinvpropdg | Unicode version |
Description: If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
grpinvpropd.1 |
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grpinvpropd.2 |
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grpinvpropdg.k |
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grpinvpropdg.l |
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grpinvpropd.3 |
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Ref | Expression |
---|---|
grpinvpropdg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvpropd.3 |
. . . . . . 7
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2 | grpinvpropd.1 |
. . . . . . . . 9
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3 | grpinvpropd.2 |
. . . . . . . . 9
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4 | grpinvpropdg.k |
. . . . . . . . 9
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5 | grpinvpropdg.l |
. . . . . . . . 9
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6 | 2, 3, 4, 5, 1 | grpidpropdg 12960 |
. . . . . . . 8
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7 | 6 | adantr 276 |
. . . . . . 7
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8 | 1, 7 | eqeq12d 2208 |
. . . . . 6
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9 | 8 | anass1rs 571 |
. . . . 5
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10 | 9 | riotabidva 5891 |
. . . 4
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11 | 10 | mpteq2dva 4120 |
. . 3
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12 | 2 | riotaeqdv 5875 |
. . . 4
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13 | 2, 12 | mpteq12dv 4112 |
. . 3
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14 | 3 | riotaeqdv 5875 |
. . . 4
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15 | 3, 14 | mpteq12dv 4112 |
. . 3
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16 | 11, 13, 15 | 3eqtr3d 2234 |
. 2
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17 | eqid 2193 |
. . . 4
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18 | eqid 2193 |
. . . 4
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19 | eqid 2193 |
. . . 4
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20 | eqid 2193 |
. . . 4
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21 | 17, 18, 19, 20 | grpinvfvalg 13117 |
. . 3
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22 | 4, 21 | syl 14 |
. 2
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23 | eqid 2193 |
. . . 4
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24 | eqid 2193 |
. . . 4
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25 | eqid 2193 |
. . . 4
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26 | eqid 2193 |
. . . 4
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27 | 23, 24, 25, 26 | grpinvfvalg 13117 |
. . 3
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28 | 5, 27 | syl 14 |
. 2
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29 | 16, 22, 28 | 3eqtr4d 2236 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-cnex 7965 ax-resscn 7966 ax-1re 7968 ax-addrcl 7971 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-inn 8985 df-ndx 12624 df-slot 12625 df-base 12627 df-0g 12872 df-minusg 13079 |
This theorem is referenced by: grpsubpropdg 13179 grpsubpropd2 13180 mulgpropdg 13237 invrpropdg 13648 rlmvnegg 13964 |
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