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Theorem expp1 10093
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 8773 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 109 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN )
3 elnnuz 9154 . . . . . . 7  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
42, 3sylib 121 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  ( ZZ>= ` 
1 ) )
5 simpll 497 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  A  e.  CC )
6 elnnuz 9154 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
7 fvconst2g 5550 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
87eleq1d 2163 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( ( NN 
X.  { A }
) `  x )  e.  CC  <->  A  e.  CC ) )
96, 8sylan2br 283 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( ( NN 
X.  { A }
) `  x )  e.  CC  <->  A  e.  CC ) )
109adantlr 462 . . . . . . 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 7566 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1312adantl 272 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
144, 11, 13seq3p1 10030 . . . . 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 8532 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
16 fvconst2g 5550 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
1715, 16sylan2 281 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
1817oveq2d 5706 . . . . 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 2127 . . . 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 10089 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 ( N  + 
1 ) ) )
2115, 20sylan2 281 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  ( N  +  1
) ) )
22 expnnval 10089 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ) `  N ) )
2322oveq1d 5705 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq 1 (  x.  ,  ( NN  X.  { A } ) ) `
 N )  x.  A ) )
2419, 21, 233eqtr4d 2137 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
25 exp1 10092 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
26 mulid2 7583 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2725, 26eqtr4d 2130 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
2827adantr 271 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
29 simpr 109 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
3029oveq1d 5705 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
31 0p1e1 8634 . . . . . 6  |-  ( 0  +  1 )  =  1
3230, 31syl6eq 2143 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
3332oveq2d 5706 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
34 oveq2 5698 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
35 exp0 10090 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3634, 35sylan9eqr 2149 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
3736oveq1d 5705 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
3828, 33, 373eqtr4d 2137 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
3924, 38jaodan 749 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
401, 39sylan2b 282 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 667    = wceq 1296    e. wcel 1445   {csn 3466    X. cxp 4465   ` cfv 5049  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448    + caddc 7450    x. cmul 7452   NNcn 8520   NN0cn0 8771   ZZ>=cuz 9118    seqcseq 10001   ^cexp 10085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-iseq 10002  df-seq3 10003  df-exp 10086
This theorem is referenced by:  expcllem  10097  expm1t  10114  expap0  10116  mulexp  10125  expadd  10128  expmul  10131  leexp2r  10140  leexp1a  10141  sqval  10144  cu2  10184  i3  10187  binom3  10202  bernneq  10205  expp1d  10218  faclbnd  10280  faclbnd2  10281  faclbnd6  10283  cjexp  10458  absexp  10643  binomlem  11042  geolim  11070  geo2sum  11073  efexp  11137  demoivreALT  11228  prmdvdsexp  11570  oddpwdclemodd  11593  expcncf  12371
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