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Theorem expp1 12142
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 10798 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 seqp1 12091 . . . . . . 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 11118 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
42, 3eleq2s 2575 . . . . . 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 466 . . . . 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 10549 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
7 fvconst2g 6115 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
86, 7sylan2 474 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
98oveq2d 6301 . . . . 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 2508 . . . 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 12138 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( N  + 
1 ) ) )
126, 11sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) ) )
13 expnnval 12138 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
1413oveq1d 6300 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
1510, 12, 143eqtr4d 2518 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
16 exp1 12141 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
17 mulid2 9595 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
1816, 17eqtr4d 2511 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
1918adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
20 simpr 461 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
2120oveq1d 6300 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
22 0p1e1 10648 . . . . . 6  |-  ( 0  +  1 )  =  1
2321, 22syl6eq 2524 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
2423oveq2d 6301 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
25 oveq2 6293 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
26 exp0 12139 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2725, 26sylan9eqr 2530 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
2827oveq1d 6300 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
2919, 24, 283eqtr4d 2518 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3015, 29jaodan 783 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
311, 30sylan2b 475 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4027    X. cxp 4997   ` cfv 5588  (class class class)co 6285   CCcc 9491   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498   NNcn 10537   NN0cn0 10796   ZZ>=cuz 11083    seqcseq 12076   ^cexp 12135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-seq 12077  df-exp 12136
This theorem is referenced by:  expcllem  12146  expm1t  12163  expeq0  12165  mulexp  12174  expadd  12177  expmul  12180  leexp2r  12192  leexp1a  12193  sqval  12196  cu2  12235  i3  12238  binom3  12256  bernneq  12261  modexp  12270  expp1d  12280  faclbnd  12337  faclbnd2  12338  faclbnd4lem1  12340  faclbnd6  12346  cjexp  12949  absexp  13103  binomlem  13607  climcndslem1  13627  climcndslem2  13628  geolim  13645  geo2sum  13648  efexp  13700  demoivreALT  13800  rpnnen2lem11  13822  prmdvdsexp  14117  pcexp  14245  prmreclem6  14301  decexp2  14423  numexpp1  14426  cnfldexp  18262  expcn  21203  mbfi1fseqlem5  21953  dvexp  22183  aaliou3lem2  22565  tangtx  22723  cxpmul2  22895  mcubic  23003  cubic2  23004  binom4  23006  dquartlem2  23008  quart1lem  23011  quart1  23012  quartlem1  23013  log2cnv  23100  log2ublem2  23103  log2ub  23105  basellem3  23181  chtublem  23311  perfectlem1  23329  perfectlem2  23330  bclbnd  23380  bposlem8  23391  dchrisum0flblem1  23518  pntlemo  23617  qabvexp  23636  rusgranumwlks  24729  nexple  27756  oddpwdc  28044  subfacval2  28382  sinccvglem  28789  heiborlem6  30142  bfplem1  30148  altgsumbcALT  32237
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