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Theorem expp1 12175
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 10818 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 seqp1 12124 . . . . . . 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 11141 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
42, 3eleq2s 2565 . . . . . 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 10568 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
7 fvconst2g 6126 . . . . . . 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 6312 . . . . 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 2498 . . . 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 12171 . . . . 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 12171 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
1413oveq1d 6311 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
1510, 12, 143eqtr4d 2508 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
16 exp1 12174 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
17 mulid2 9611 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
1816, 17eqtr4d 2501 . . . . 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 6311 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
22 0p1e1 10668 . . . . . 6  |-  ( 0  +  1 )  =  1
2321, 22syl6eq 2514 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
2423oveq2d 6312 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
25 oveq2 6304 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
26 exp0 12172 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2725, 26sylan9eqr 2520 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
2827oveq1d 6311 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
2919, 24, 283eqtr4d 2508 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3015, 29jaodan 785 . 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 1395    e. wcel 1819   {csn 4032    X. cxp 5006   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NNcn 10556   NN0cn0 10816   ZZ>=cuz 11106    seqcseq 12109   ^cexp 12168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12110  df-exp 12169
This theorem is referenced by:  expcllem  12179  expm1t  12196  expeq0  12198  mulexp  12207  expadd  12210  expmul  12213  leexp2r  12225  leexp1a  12226  sqval  12229  cu2  12268  i3  12271  binom3  12289  bernneq  12294  modexp  12303  expp1d  12313  faclbnd  12370  faclbnd2  12371  faclbnd4lem1  12373  faclbnd6  12379  cjexp  12994  absexp  13148  binomlem  13652  climcndslem1  13672  climcndslem2  13673  geolim  13690  geo2sum  13693  efexp  13847  demoivreALT  13947  rpnnen2lem11  13969  prmdvdsexp  14266  pcexp  14394  prmreclem6  14450  decexp2  14572  numexpp1  14575  cnfldexp  18577  expcn  21501  mbfi1fseqlem5  22251  dvexp  22481  aaliou3lem2  22864  tangtx  23023  cxpmul2  23195  mcubic  23303  cubic2  23304  binom4  23306  dquartlem2  23308  quart1lem  23311  quart1  23312  quartlem1  23313  log2cnv  23400  log2ublem2  23403  log2ub  23405  basellem3  23481  chtublem  23611  perfectlem1  23629  perfectlem2  23630  bclbnd  23680  bposlem8  23691  dchrisum0flblem1  23818  pntlemo  23917  qabvexp  23936  rusgranumwlks  25082  oddpwdc  28468  subfacval2  28806  sinccvglem  29213  heiborlem6  30474  bfplem1  30480  altgsumbcALT  33044
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