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Mirrors > Home > ILE Home > Th. List > grppropd | Unicode version |
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
grppropd.1 |
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grppropd.2 |
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grppropd.3 |
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Ref | Expression |
---|---|
grppropd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grppropd.1 |
. . . 4
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2 | grppropd.2 |
. . . 4
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3 | grppropd.3 |
. . . 4
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4 | 1, 2, 3 | mndpropd 12867 |
. . 3
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5 | 1 | adantr 276 |
. . . . . . . . 9
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6 | 2 | adantr 276 |
. . . . . . . . 9
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7 | simprl 529 |
. . . . . . . . . 10
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8 | 5, 7 | basmexd 12540 |
. . . . . . . . 9
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9 | 6, 7 | basmexd 12540 |
. . . . . . . . 9
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10 | 3 | ralrimivva 2572 |
. . . . . . . . . . . . . 14
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11 | oveq1 5898 |
. . . . . . . . . . . . . . . 16
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12 | oveq1 5898 |
. . . . . . . . . . . . . . . 16
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13 | 11, 12 | eqeq12d 2204 |
. . . . . . . . . . . . . . 15
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14 | oveq2 5899 |
. . . . . . . . . . . . . . . 16
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15 | oveq2 5899 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | eqeq12d 2204 |
. . . . . . . . . . . . . . 15
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17 | 13, 16 | cbvral2v 2731 |
. . . . . . . . . . . . . 14
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18 | 10, 17 | sylib 122 |
. . . . . . . . . . . . 13
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19 | 18 | adantr 276 |
. . . . . . . . . . . 12
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20 | 19 | r19.21bi 2578 |
. . . . . . . . . . 11
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21 | 20 | r19.21bi 2578 |
. . . . . . . . . 10
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22 | 21 | anasss 399 |
. . . . . . . . 9
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23 | 5, 6, 8, 9, 22 | grpidpropdg 12816 |
. . . . . . . 8
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24 | 3, 23 | eqeq12d 2204 |
. . . . . . 7
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25 | 24 | anass1rs 571 |
. . . . . 6
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26 | 25 | rexbidva 2487 |
. . . . 5
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27 | 26 | ralbidva 2486 |
. . . 4
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28 | 1 | rexeqdv 2693 |
. . . . 5
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29 | 1, 28 | raleqbidv 2698 |
. . . 4
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30 | 2 | rexeqdv 2693 |
. . . . 5
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31 | 2, 30 | raleqbidv 2698 |
. . . 4
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32 | 27, 29, 31 | 3bitr3d 218 |
. . 3
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33 | 4, 32 | anbi12d 473 |
. 2
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34 | eqid 2189 |
. . 3
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35 | eqid 2189 |
. . 3
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36 | eqid 2189 |
. . 3
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37 | 34, 35, 36 | isgrp 12917 |
. 2
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38 | eqid 2189 |
. . 3
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39 | eqid 2189 |
. . 3
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40 | eqid 2189 |
. . 3
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41 | 38, 39, 40 | isgrp 12917 |
. 2
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42 | 33, 37, 41 | 3bitr4g 223 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-riota 5847 df-ov 5894 df-inn 8938 df-2 8996 df-ndx 12483 df-slot 12484 df-base 12486 df-plusg 12568 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 |
This theorem is referenced by: grpprop 12929 grppropstrg 12930 ghmpropd 13183 ablpropd 13196 ringpropd 13353 opprring 13390 opprsubgg 13395 lmodprop2d 13625 sralmod 13727 |
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