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Theorem expp1 11076
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 9934 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 seqp1 11027 . . . . . . 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 10230 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
42, 3eleq2s 2350 . . . . . 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 454 . . . . 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 9726 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
7 fvconst2g 5661 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
86, 7sylan2 462 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
98oveq2d 5808 . . . . 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 2290 . . . 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 11073 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( N  + 
1 ) ) )
126, 11sylan2 462 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) ) )
13 expnnval 11073 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq  1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
1413oveq1d 5807 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq  1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
1510, 12, 143eqtr4d 2300 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
16 exp1 11075 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
17 mulid2 8803 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
1816, 17eqtr4d 2293 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
1918adantr 453 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
20 simpr 449 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
2120oveq1d 5807 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
22 0p1e1 9807 . . . . . 6  |-  ( 0  +  1 )  =  1
2321, 22syl6eq 2306 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
2423oveq2d 5808 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
25 oveq2 5800 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
26 exp0 11074 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2725, 26sylan9eqr 2312 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
2827oveq1d 5807 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
2919, 24, 283eqtr4d 2300 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3015, 29jaodan 763 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
311, 30sylan2b 463 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   {csn 3614    X. cxp 4659   ` cfv 4673  (class class class)co 5792   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710   NNcn 9714   NN0cn0 9932   ZZ>=cuz 10197    seq cseq 11012   ^cexp 11070
This theorem is referenced by:  expcllem  11080  expm1t  11096  expeq0  11098  mulexp  11107  expadd  11110  expmul  11113  leexp2r  11125  leexp1a  11126  sqval  11129  cu2  11167  i3  11170  binom3  11188  bernneq  11193  modexp  11202  expp1d  11212  faclbnd  11269  faclbnd2  11270  faclbnd4lem1  11272  faclbnd6  11278  cjexp  11600  absexp  11754  binomlem  12252  climcndslem1  12270  climcndslem2  12271  geolim  12288  geo2sum  12291  efexp  12343  demoivreALT  12443  rpnnen2lem11  12465  prmdvdsexp  12755  pcexp  12874  prmreclem6  12930  decexp2  13052  numexpp1  13055  cnfldexp  16369  expcn  18338  mbfi1fseqlem5  19036  dvexp  19264  aaliou3lem2  19685  tangtx  19835  cxpmul2  19998  mcubic  20105  cubic2  20106  binom4  20108  dquartlem2  20110  quart1lem  20113  quart1  20114  quartlem1  20115  log2cnv  20202  log2ublem2  20205  log2ub  20207  basellem3  20282  chtublem  20412  perfectlem1  20430  perfectlem2  20431  bclbnd  20481  bposlem8  20492  dchrisum0flblem1  20619  pntlemo  20718  qabvexp  20737  subfacval2  23090  sinccvglem  23377  heiborlem6  25907  bfplem1  25913  stoweidlem3  27087  stoweidlem19  27103
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-n0 9933  df-z 9992  df-uz 10198  df-seq 11013  df-exp 11071
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