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Theorem expp1 11110
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 9967 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 seqp1 11061 . . . . . . 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 10263 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
42, 3eleq2s 2375 . . . . . 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 452 . . . . 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 9758 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
7 fvconst2g 5727 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
86, 7sylan2 460 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
98oveq2d 5874 . . . . 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 2315 . . . 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 11107 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( N  + 
1 ) ) )
126, 11sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) ) )
13 expnnval 11107 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
1413oveq1d 5873 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
1510, 12, 143eqtr4d 2325 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
16 exp1 11109 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
17 mulid2 8836 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
1816, 17eqtr4d 2318 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
1918adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
20 simpr 447 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
2120oveq1d 5873 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
22 0p1e1 9839 . . . . . 6  |-  ( 0  +  1 )  =  1
2321, 22syl6eq 2331 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
2423oveq2d 5874 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
25 oveq2 5866 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
26 exp0 11108 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2725, 26sylan9eqr 2337 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
2827oveq1d 5873 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
2919, 24, 283eqtr4d 2325 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3015, 29jaodan 760 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
311, 30sylan2b 461 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 357    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   NN0cn0 9965   ZZ>=cuz 10230    seq cseq 11046   ^cexp 11104
This theorem is referenced by:  expcllem  11114  expm1t  11130  expeq0  11132  mulexp  11141  expadd  11144  expmul  11147  leexp2r  11159  leexp1a  11160  sqval  11163  cu2  11201  i3  11204  binom3  11222  bernneq  11227  modexp  11236  expp1d  11246  faclbnd  11303  faclbnd2  11304  faclbnd4lem1  11306  faclbnd6  11312  cjexp  11635  absexp  11789  binomlem  12287  climcndslem1  12308  climcndslem2  12309  geolim  12326  geo2sum  12329  efexp  12381  demoivreALT  12481  rpnnen2lem11  12503  prmdvdsexp  12793  pcexp  12912  prmreclem6  12968  decexp2  13090  numexpp1  13093  cnfldexp  16407  expcn  18376  mbfi1fseqlem5  19074  dvexp  19302  aaliou3lem2  19723  tangtx  19873  cxpmul2  20036  mcubic  20143  cubic2  20144  binom4  20146  dquartlem2  20148  quart1lem  20151  quart1  20152  quartlem1  20153  log2cnv  20240  log2ublem2  20243  log2ub  20245  basellem3  20320  chtublem  20450  perfectlem1  20468  perfectlem2  20469  bclbnd  20519  bposlem8  20530  dchrisum0flblem1  20657  pntlemo  20756  qabvexp  20775  subfacval2  23718  sinccvglem  24005  heiborlem6  26540  bfplem1  26546  stoweidlem3  27752  stoweidlem19  27768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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