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Mirrors > Home > ILE Home > Th. List > mulgneg | Unicode version |
Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnncl.b |
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mulgnncl.t |
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mulgneg.i |
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Ref | Expression |
---|---|
mulgneg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 9154 |
. . 3
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2 | simpr 110 |
. . . . 5
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3 | simpl3 1002 |
. . . . 5
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4 | mulgnncl.b |
. . . . . 6
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5 | mulgnncl.t |
. . . . . 6
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6 | mulgneg.i |
. . . . . 6
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7 | 4, 5, 6 | mulgnegnn 12869 |
. . . . 5
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8 | 2, 3, 7 | syl2anc 411 |
. . . 4
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9 | simpl1 1000 |
. . . . . 6
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10 | eqid 2177 |
. . . . . . 7
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11 | 10, 6 | grpinvid 12807 |
. . . . . 6
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12 | 9, 11 | syl 14 |
. . . . 5
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13 | simpr 110 |
. . . . . . . 8
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14 | 13 | oveq1d 5883 |
. . . . . . 7
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15 | simpl3 1002 |
. . . . . . . 8
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16 | 4, 10, 5 | mulg0 12864 |
. . . . . . . 8
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17 | 15, 16 | syl 14 |
. . . . . . 7
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18 | 14, 17 | eqtrd 2210 |
. . . . . 6
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19 | 18 | fveq2d 5514 |
. . . . 5
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20 | 13 | negeqd 8129 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | neg0 8180 |
. . . . . . . 8
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22 | 20, 21 | eqtrdi 2226 |
. . . . . . 7
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23 | 22 | oveq1d 5883 |
. . . . . 6
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24 | 23, 17 | eqtrd 2210 |
. . . . 5
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25 | 12, 19, 24 | 3eqtr4rd 2221 |
. . . 4
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26 | 8, 25 | jaodan 797 |
. . 3
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27 | 1, 26 | sylan2b 287 |
. 2
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28 | simpl1 1000 |
. . . 4
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29 | simprr 531 |
. . . . . 6
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30 | 29 | nnzd 9350 |
. . . . 5
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31 | simpl3 1002 |
. . . . 5
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32 | 4, 5 | mulgcl 12876 |
. . . . 5
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33 | 28, 30, 31, 32 | syl3anc 1238 |
. . . 4
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34 | 4, 6 | grpinvinv 12813 |
. . . 4
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35 | 28, 33, 34 | syl2anc 411 |
. . 3
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36 | 4, 5, 6 | mulgnegnn 12869 |
. . . . . 6
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37 | 29, 31, 36 | syl2anc 411 |
. . . . 5
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38 | simprl 529 |
. . . . . . . 8
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39 | 38 | recnd 7963 |
. . . . . . 7
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40 | 39 | negnegd 8236 |
. . . . . 6
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41 | 40 | oveq1d 5883 |
. . . . 5
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42 | 37, 41 | eqtr3d 2212 |
. . . 4
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43 | 42 | fveq2d 5514 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 35, 43 | eqtr3d 2212 |
. 2
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45 | simp2 998 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
46 | elznn0nn 9243 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 45, 46 | sylib 122 |
. 2
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48 | 27, 44, 47 | mpjaodan 798 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 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-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 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-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-n0 9153 df-z 9230 df-uz 9505 df-seqfrec 10419 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 df-minusg 12758 df-mulg 12860 |
This theorem is referenced by: mulgnegneg 12878 mulgm1 12879 mulgaddcomlem 12881 mulginvcom 12883 mulgz 12886 mulgdirlem 12889 mulgdir 12890 mulgneg2 12892 mulgass 12895 mulgsubdir 12898 mulgass2 13048 |
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