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Theorem expp1 10293
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 8972 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 109 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN )
3 elnnuz 9355 . . . . . . 7  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
42, 3sylib 121 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  ( ZZ>= ` 
1 ) )
5 simpll 518 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  A  e.  CC )
6 elnnuz 9355 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
7 fvconst2g 5627 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
87eleq1d 2206 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( ( NN 
X.  { A }
) `  x )  e.  CC  <->  A  e.  CC ) )
96, 8sylan2br 286 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( ( NN 
X.  { A }
) `  x )  e.  CC  <->  A  e.  CC ) )
109adantlr 468 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  ( ( ( NN  X.  { A } ) `  x
)  e.  CC  <->  A  e.  CC ) )
115, 10mpbird 166 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  ( ( NN 
X.  { A }
) `  x )  e.  CC )
12 mulcl 7740 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1312adantl 275 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
144, 11, 13seq3p1 10228 . . . . 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
) ) ) )
15 peano2nn 8725 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
16 fvconst2g 5627 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
1715, 16sylan2 284 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
1817oveq2d 5783 . . . . 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 ) )
1914, 18eqtrd 2170 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ) `  ( N  +  1 ) )  =  ( (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
20 expnnval 10289 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( N  + 
1 ) ) )
2115, 20sylan2 284 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) ) )
22 expnnval 10289 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2322oveq1d 5782 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
2419, 21, 233eqtr4d 2180 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
25 exp1 10292 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
26 mulid2 7757 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2725, 26eqtr4d 2173 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
2827adantr 274 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
29 simpr 109 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
3029oveq1d 5782 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
31 0p1e1 8827 . . . . . 6  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2186 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
3332oveq2d 5783 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
34 oveq2 5775 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
35 exp0 10290 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3634, 35sylan9eqr 2192 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
3736oveq1d 5782 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
3828, 33, 373eqtr4d 2180 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3924, 38jaodan 786 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
401, 39sylan2b 285 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   {csn 3522    X. cxp 4532   ` cfv 5118  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618   NNcn 8713   NN0cn0 8970   ZZ>=cuz 9319    seqcseq 10211   ^cexp 10285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-seqfrec 10212  df-exp 10286
This theorem is referenced by:  expcllem  10297  expm1t  10314  expap0  10316  mulexp  10325  expadd  10328  expmul  10331  leexp2r  10340  leexp1a  10341  sqval  10344  cu2  10384  i3  10387  binom3  10402  bernneq  10405  expp1d  10418  faclbnd  10480  faclbnd2  10481  faclbnd6  10483  cjexp  10658  absexp  10844  binomlem  11245  geolim  11273  geo2sum  11276  efexp  11377  demoivreALT  11469  prmdvdsexp  11815  oddpwdclemodd  11839  expcncf  12750  dvexp  12833  tangtx  12908
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