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Mirrors > Home > ILE Home > Th. List > expp1 | Unicode version |
Description: Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.) |
Ref | Expression |
---|---|
expp1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 9003 |
. 2
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2 | simpr 109 |
. . . . . . 7
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3 | elnnuz 9386 |
. . . . . . 7
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4 | 2, 3 | sylib 121 |
. . . . . 6
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5 | simpll 519 |
. . . . . . 7
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6 | elnnuz 9386 |
. . . . . . . . 9
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7 | fvconst2g 5642 |
. . . . . . . . . 10
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8 | 7 | eleq1d 2209 |
. . . . . . . . 9
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9 | 6, 8 | sylan2br 286 |
. . . . . . . 8
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10 | 9 | adantlr 469 |
. . . . . . 7
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11 | 5, 10 | mpbird 166 |
. . . . . 6
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12 | mulcl 7771 |
. . . . . . 7
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13 | 12 | adantl 275 |
. . . . . 6
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14 | 4, 11, 13 | seq3p1 10266 |
. . . . 5
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15 | peano2nn 8756 |
. . . . . . 7
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16 | fvconst2g 5642 |
. . . . . . 7
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17 | 15, 16 | sylan2 284 |
. . . . . 6
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18 | 17 | oveq2d 5798 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 14, 18 | eqtrd 2173 |
. . . 4
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20 | expnnval 10327 |
. . . . 5
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21 | 15, 20 | sylan2 284 |
. . . 4
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22 | expnnval 10327 |
. . . . 5
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23 | 22 | oveq1d 5797 |
. . . 4
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24 | 19, 21, 23 | 3eqtr4d 2183 |
. . 3
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25 | exp1 10330 |
. . . . . 6
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26 | mulid2 7788 |
. . . . . 6
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27 | 25, 26 | eqtr4d 2176 |
. . . . 5
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28 | 27 | adantr 274 |
. . . 4
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29 | simpr 109 |
. . . . . . 7
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30 | 29 | oveq1d 5797 |
. . . . . 6
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31 | 0p1e1 8858 |
. . . . . 6
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32 | 30, 31 | eqtrdi 2189 |
. . . . 5
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33 | 32 | oveq2d 5798 |
. . . 4
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34 | oveq2 5790 |
. . . . . 6
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35 | exp0 10328 |
. . . . . 6
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36 | 34, 35 | sylan9eqr 2195 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | oveq1d 5797 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 28, 33, 37 | 3eqtr4d 2183 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 24, 38 | jaodan 787 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 1, 39 | sylan2b 285 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-seqfrec 10250 df-exp 10324 |
This theorem is referenced by: expcllem 10335 expm1t 10352 expap0 10354 mulexp 10363 expadd 10366 expmul 10369 leexp2r 10378 leexp1a 10379 sqval 10382 cu2 10422 i3 10425 binom3 10440 bernneq 10443 expp1d 10456 faclbnd 10519 faclbnd2 10520 faclbnd6 10522 cjexp 10697 absexp 10883 binomlem 11284 geolim 11312 geo2sum 11315 efexp 11425 demoivreALT 11516 prmdvdsexp 11862 oddpwdclemodd 11886 expcncf 12800 dvexp 12883 tangtx 12967 |
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