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Mirrors > Home > ILE Home > Th. List > grp1 | Unicode version |
Description: The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.) |
Ref | Expression |
---|---|
grp1.m |
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Ref | Expression |
---|---|
grp1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grp1.m |
. . 3
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2 | 1 | mnd1 12724 |
. 2
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3 | df-ov 5871 |
. . . . 5
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4 | opexg 4224 |
. . . . . . 7
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5 | 4 | anidms 397 |
. . . . . 6
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6 | fvsng 5707 |
. . . . . 6
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7 | 5, 6 | mpancom 422 |
. . . . 5
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8 | 3, 7 | eqtrid 2222 |
. . . 4
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9 | 1 | mnd1id 12725 |
. . . 4
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10 | 8, 9 | eqtr4d 2213 |
. . 3
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11 | oveq2 5876 |
. . . . . . 7
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12 | 11 | eqeq1d 2186 |
. . . . . 6
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13 | 12 | rexbidv 2478 |
. . . . 5
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14 | 13 | ralsng 3631 |
. . . 4
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15 | oveq1 5875 |
. . . . . 6
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16 | 15 | eqeq1d 2186 |
. . . . 5
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17 | 16 | rexsng 3632 |
. . . 4
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18 | 14, 17 | bitrd 188 |
. . 3
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19 | 10, 18 | mpbird 167 |
. 2
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20 | eqid 2177 |
. . . 4
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21 | eqid 2177 |
. . . 4
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22 | eqid 2177 |
. . . 4
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23 | 20, 21, 22 | isgrp 12760 |
. . 3
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24 | snexg 4181 |
. . . . . 6
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25 | opexg 4224 |
. . . . . . . 8
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26 | 5, 25 | mpancom 422 |
. . . . . . 7
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27 | snexg 4181 |
. . . . . . 7
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28 | 26, 27 | syl 14 |
. . . . . 6
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29 | 1 | grpbaseg 12551 |
. . . . . 6
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30 | 24, 28, 29 | syl2anc 411 |
. . . . 5
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31 | 1 | grpplusgg 12552 |
. . . . . . . . 9
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32 | 24, 28, 31 | syl2anc 411 |
. . . . . . . 8
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33 | 32 | oveqd 5885 |
. . . . . . 7
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34 | 33 | eqeq1d 2186 |
. . . . . 6
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35 | 30, 34 | rexeqbidv 2685 |
. . . . 5
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36 | 30, 35 | raleqbidv 2684 |
. . . 4
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37 | 36 | anbi2d 464 |
. . 3
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38 | 23, 37 | bitr4id 199 |
. 2
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39 | 2, 19, 38 | mpbir2and 944 |
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-in1 614 ax-in2 615 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-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 |
This theorem is referenced by: grp1inv 12853 ring1 13049 |
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