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Mirrors > Home > ILE Home > Th. List > grpsubpropd2 | Unicode version |
Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
grpsubpropd2.1 |
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grpsubpropd2.2 |
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grpsubpropd2.3 |
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grpsubpropd2.4 |
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Ref | Expression |
---|---|
grpsubpropd2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 |
. . . . . 6
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2 | simp2 998 |
. . . . . . 7
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3 | grpsubpropd2.1 |
. . . . . . . 8
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4 | 3 | 3ad2ant1 1018 |
. . . . . . 7
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5 | 2, 4 | eleqtrrd 2257 |
. . . . . 6
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6 | grpsubpropd2.3 |
. . . . . . . . 9
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7 | 6 | 3ad2ant1 1018 |
. . . . . . . 8
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8 | simp3 999 |
. . . . . . . 8
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9 | eqid 2177 |
. . . . . . . . 9
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10 | eqid 2177 |
. . . . . . . . 9
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11 | 9, 10 | grpinvcl 12926 |
. . . . . . . 8
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12 | 7, 8, 11 | syl2anc 411 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 12, 4 | eleqtrrd 2257 |
. . . . . 6
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14 | grpsubpropd2.4 |
. . . . . . 7
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15 | 14 | oveqrspc2v 5904 |
. . . . . 6
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16 | 1, 5, 13, 15 | syl12anc 1236 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | grpsubpropd2.2 |
. . . . . . . . 9
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18 | eqid 2177 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 9, 18 | grpidcl 12909 |
. . . . . . . . . . . 12
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20 | 6, 19 | syl 14 |
. . . . . . . . . . 11
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21 | 20, 3 | eleqtrrd 2257 |
. . . . . . . . . 10
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22 | 17, 21 | basmexd 12524 |
. . . . . . . . 9
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23 | 3, 17, 6, 22, 14 | grpinvpropdg 12950 |
. . . . . . . 8
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24 | 23 | fveq1d 5519 |
. . . . . . 7
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25 | 24 | oveq2d 5893 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | 3ad2ant1 1018 |
. . . . 5
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27 | 16, 26 | eqtrd 2210 |
. . . 4
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28 | 27 | mpoeq3dva 5941 |
. . 3
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29 | 3, 17 | eqtr3d 2212 |
. . . 4
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30 | mpoeq12 5937 |
. . . 4
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31 | 29, 29, 30 | syl2anc 411 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 28, 31 | eqtrd 2210 |
. 2
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33 | eqid 2177 |
. . . 4
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34 | eqid 2177 |
. . . 4
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35 | 9, 33, 10, 34 | grpsubfvalg 12923 |
. . 3
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36 | 6, 35 | syl 14 |
. 2
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37 | eqid 2177 |
. . . 4
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38 | eqid 2177 |
. . . 4
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39 | eqid 2177 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | eqid 2177 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | 37, 38, 39, 40 | grpsubfvalg 12923 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 22, 41 | syl 14 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 32, 36, 42 | 3eqtr4d 2220 |
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 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-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-inn 8922 df-2 8980 df-ndx 12467 df-slot 12468 df-base 12470 df-plusg 12551 df-0g 12712 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-grp 12885 df-minusg 12886 df-sbg 12887 |
This theorem is referenced by: (None) |
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