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Theorem expp1 11347
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.)
Assertion
Ref Expression
expp1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )

Proof of Theorem expp1
StepHypRef Expression
1 elnn0 10183 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 seqp1 11297 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  1
)  ->  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) )  =  ( (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
)  x.  ( ( NN  X.  { A } ) `  ( N  +  1 ) ) ) )
3 nnuz 10481 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
42, 3eleq2s 2500 . . . . . 6  |-  ( N  e.  NN  ->  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) )  =  ( (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
)  x.  ( ( NN  X.  { A } ) `  ( N  +  1 ) ) ) )
54adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  ( N  +  1 ) )  =  ( (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  ( ( NN  X.  { A } ) `  ( N  +  1
) ) ) )
6 peano2nn 9972 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
7 fvconst2g 5908 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
86, 7sylan2 461 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
98oveq2d 6060 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( (  seq  1
(  x.  ,  ( NN  X.  { A } ) ) `  N )  x.  (
( NN  X.  { A } ) `  ( N  +  1 ) ) )  =  ( (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  N
)  x.  A ) )
105, 9eqtrd 2440 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq  1 (  x.  ,  ( NN 
X.  { A }
) ) `  ( N  +  1 ) )  =  ( (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
11 expnnval 11344 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( N  + 
1 ) ) )
126, 11sylan2 461 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) ) )
13 expnnval 11344 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
1413oveq1d 6059 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
1510, 12, 143eqtr4d 2450 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
16 exp1 11346 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
17 mulid2 9049 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
1816, 17eqtr4d 2443 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
1918adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
20 simpr 448 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
2120oveq1d 6059 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
22 0p1e1 10053 . . . . . 6  |-  ( 0  +  1 )  =  1
2321, 22syl6eq 2456 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
2423oveq2d 6060 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
25 oveq2 6052 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
26 exp0 11345 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2725, 26sylan9eqr 2462 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
2827oveq1d 6059 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
2919, 24, 283eqtr4d 2450 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3015, 29jaodan 761 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
311, 30sylan2b 462 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3778    X. cxp 4839   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953    x. cmul 8955   NNcn 9960   NN0cn0 10181   ZZ>=cuz 10448    seq cseq 11282   ^cexp 11341
This theorem is referenced by:  expcllem  11351  expm1t  11367  expeq0  11369  mulexp  11378  expadd  11381  expmul  11384  leexp2r  11396  leexp1a  11397  sqval  11400  cu2  11438  i3  11441  binom3  11459  bernneq  11464  modexp  11473  expp1d  11483  faclbnd  11540  faclbnd2  11541  faclbnd4lem1  11543  faclbnd6  11549  cjexp  11914  absexp  12068  binomlem  12567  climcndslem1  12588  climcndslem2  12589  geolim  12606  geo2sum  12609  efexp  12661  demoivreALT  12761  rpnnen2lem11  12783  prmdvdsexp  13073  pcexp  13192  prmreclem6  13248  decexp2  13370  numexpp1  13373  cnfldexp  16693  expcn  18859  mbfi1fseqlem5  19568  dvexp  19796  aaliou3lem2  20217  tangtx  20370  cxpmul2  20537  mcubic  20644  cubic2  20645  binom4  20647  dquartlem2  20649  quart1lem  20652  quart1  20653  quartlem1  20654  log2cnv  20741  log2ublem2  20744  log2ub  20746  basellem3  20822  chtublem  20952  perfectlem1  20970  perfectlem2  20971  bclbnd  21021  bposlem8  21032  dchrisum0flblem1  21159  pntlemo  21258  qabvexp  21277  subfacval2  24830  sinccvglem  25066  heiborlem6  26419  bfplem1  26425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-seq 11283  df-exp 11342
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