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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 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnn0 9297 |
. 2
| |
| 2 | simpr 110 |
. . . . . . 7
| |
| 3 | elnnuz 9685 |
. . . . . . 7
| |
| 4 | 2, 3 | sylib 122 |
. . . . . 6
|
| 5 | simpll 527 |
. . . . . . 7
| |
| 6 | elnnuz 9685 |
. . . . . . . . 9
| |
| 7 | fvconst2g 5798 |
. . . . . . . . . 10
| |
| 8 | 7 | eleq1d 2274 |
. . . . . . . . 9
|
| 9 | 6, 8 | sylan2br 288 |
. . . . . . . 8
|
| 10 | 9 | adantlr 477 |
. . . . . . 7
|
| 11 | 5, 10 | mpbird 167 |
. . . . . 6
|
| 12 | mulcl 8052 |
. . . . . . 7
| |
| 13 | 12 | adantl 277 |
. . . . . 6
|
| 14 | 4, 11, 13 | seq3p1 10610 |
. . . . 5
|
| 15 | peano2nn 9048 |
. . . . . . 7
| |
| 16 | fvconst2g 5798 |
. . . . . . 7
| |
| 17 | 15, 16 | sylan2 286 |
. . . . . 6
|
| 18 | 17 | oveq2d 5960 |
. . . . 5
|
| 19 | 14, 18 | eqtrd 2238 |
. . . 4
|
| 20 | expnnval 10687 |
. . . . 5
| |
| 21 | 15, 20 | sylan2 286 |
. . . 4
|
| 22 | expnnval 10687 |
. . . . 5
| |
| 23 | 22 | oveq1d 5959 |
. . . 4
|
| 24 | 19, 21, 23 | 3eqtr4d 2248 |
. . 3
|
| 25 | exp1 10690 |
. . . . . 6
| |
| 26 | mullid 8070 |
. . . . . 6
| |
| 27 | 25, 26 | eqtr4d 2241 |
. . . . 5
|
| 28 | 27 | adantr 276 |
. . . 4
|
| 29 | simpr 110 |
. . . . . . 7
| |
| 30 | 29 | oveq1d 5959 |
. . . . . 6
|
| 31 | 0p1e1 9150 |
. . . . . 6
| |
| 32 | 30, 31 | eqtrdi 2254 |
. . . . 5
|
| 33 | 32 | oveq2d 5960 |
. . . 4
|
| 34 | oveq2 5952 |
. . . . . 6
| |
| 35 | exp0 10688 |
. . . . . 6
| |
| 36 | 34, 35 | sylan9eqr 2260 |
. . . . 5
|
| 37 | 36 | oveq1d 5959 |
. . . 4
|
| 38 | 28, 33, 37 | 3eqtr4d 2248 |
. . 3
|
| 39 | 24, 38 | jaodan 799 |
. 2
|
| 40 | 1, 39 | sylan2b 287 |
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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 |
| This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-seqfrec 10593 df-exp 10684 |
| This theorem is referenced by: expcllem 10695 expm1t 10712 expap0 10714 mulexp 10723 expadd 10726 expmul 10729 leexp2r 10738 leexp1a 10739 sqval 10742 cu2 10783 i3 10786 binom3 10802 bernneq 10805 modqexp 10811 expp1d 10819 faclbnd 10886 faclbnd2 10887 faclbnd6 10889 cjexp 11204 absexp 11390 binomlem 11794 geolim 11822 geo2sum 11825 efexp 11993 demoivreALT 12085 prmdvdsexp 12470 oddpwdclemodd 12494 pcexp 12632 numexpp1 12747 2exp7 12757 cnfldexp 14339 expcn 15041 expcncf 15081 dvexp 15183 tangtx 15310 rpcxpmul2 15385 binom4 15451 perfectlem1 15471 perfectlem2 15472 |
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