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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 9242 |
. 2
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2 | simpr 110 |
. . . . . . 7
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3 | elnnuz 9629 |
. . . . . . 7
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4 | 2, 3 | sylib 122 |
. . . . . 6
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5 | simpll 527 |
. . . . . . 7
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6 | elnnuz 9629 |
. . . . . . . . 9
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7 | fvconst2g 5772 |
. . . . . . . . . 10
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8 | 7 | eleq1d 2262 |
. . . . . . . . 9
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9 | 6, 8 | sylan2br 288 |
. . . . . . . 8
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10 | 9 | adantlr 477 |
. . . . . . 7
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11 | 5, 10 | mpbird 167 |
. . . . . 6
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12 | mulcl 7999 |
. . . . . . 7
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13 | 12 | adantl 277 |
. . . . . 6
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14 | 4, 11, 13 | seq3p1 10536 |
. . . . 5
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15 | peano2nn 8994 |
. . . . . . 7
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16 | fvconst2g 5772 |
. . . . . . 7
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17 | 15, 16 | sylan2 286 |
. . . . . 6
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18 | 17 | oveq2d 5934 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 14, 18 | eqtrd 2226 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | expnnval 10613 |
. . . . 5
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21 | 15, 20 | sylan2 286 |
. . . 4
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22 | expnnval 10613 |
. . . . 5
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23 | 22 | oveq1d 5933 |
. . . 4
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24 | 19, 21, 23 | 3eqtr4d 2236 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | exp1 10616 |
. . . . . 6
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26 | mullid 8017 |
. . . . . 6
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27 | 25, 26 | eqtr4d 2229 |
. . . . 5
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28 | 27 | adantr 276 |
. . . 4
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29 | simpr 110 |
. . . . . . 7
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30 | 29 | oveq1d 5933 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 0p1e1 9096 |
. . . . . 6
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32 | 30, 31 | eqtrdi 2242 |
. . . . 5
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33 | 32 | oveq2d 5934 |
. . . 4
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34 | oveq2 5926 |
. . . . . 6
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35 | exp0 10614 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 34, 35 | sylan9eqr 2248 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 36 | oveq1d 5933 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 28, 33, 37 | 3eqtr4d 2236 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 24, 38 | jaodan 798 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 1, 39 | sylan2b 287 |
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 615 ax-in2 616 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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-seqfrec 10519 df-exp 10610 |
This theorem is referenced by: expcllem 10621 expm1t 10638 expap0 10640 mulexp 10649 expadd 10652 expmul 10655 leexp2r 10664 leexp1a 10665 sqval 10668 cu2 10709 i3 10712 binom3 10728 bernneq 10731 modqexp 10737 expp1d 10745 faclbnd 10812 faclbnd2 10813 faclbnd6 10815 cjexp 11037 absexp 11223 binomlem 11626 geolim 11654 geo2sum 11657 efexp 11825 demoivreALT 11917 prmdvdsexp 12286 oddpwdclemodd 12310 pcexp 12447 cnfldexp 14065 expcncf 14763 dvexp 14860 tangtx 14973 binom4 15111 |
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