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Theorem expp1 9427
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 8241 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 107 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  NN )
3 elnnuz 8605 . . . . . . 7  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
42, 3sylib 131 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  N  e.  ( ZZ>= ` 
1 ) )
5 cnex 7063 . . . . . . 7  |-  CC  e.  _V
65a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  CC  e.  _V )
7 simpll 489 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  A  e.  CC )
8 elnnuz 8605 . . . . . . . . 9  |-  ( x  e.  NN  <->  x  e.  ( ZZ>= `  1 )
)
9 fvconst2g 5403 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( NN  X.  { A } ) `  x )  =  A )
109eleq1d 2122 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  NN )  ->  ( ( ( NN 
X.  { A }
) `  x )  e.  CC  <->  A  e.  CC ) )
118, 10sylan2br 276 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  ( ZZ>= ` 
1 ) )  -> 
( ( ( NN 
X.  { A }
) `  x )  e.  CC  <->  A  e.  CC ) )
1211adantlr 454 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  ( ( ( NN  X.  { A } ) `  x
)  e.  CC  <->  A  e.  CC ) )
137, 12mpbird 160 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  x  e.  (
ZZ>= `  1 ) )  ->  ( ( NN 
X.  { A }
) `  x )  e.  CC )
14 mulcl 7066 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1514adantl 266 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN )  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
164, 6, 13, 15iseqp1 9389 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 ( N  + 
1 ) )  =  ( (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )  x.  ( ( NN  X.  { A } ) `  ( N  +  1
) ) ) )
17 peano2nn 8002 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
18 fvconst2g 5403 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( ( NN 
X.  { A }
) `  ( N  +  1 ) )  =  A )
1917, 18sylan2 274 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( NN  X.  { A } ) `  ( N  +  1
) )  =  A )
2019oveq2d 5556 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )  x.  ( ( NN  X.  { A } ) `  ( N  +  1
) ) )  =  ( (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )  x.  A ) )
2116, 20eqtrd 2088 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  (  seq 1 (  x.  ,  ( NN 
X.  { A }
) ,  CC ) `
 ( N  + 
1 ) )  =  ( (  seq 1
(  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N )  x.  A ) )
22 expinnval 9423 . . . . 5  |-  ( ( A  e.  CC  /\  ( N  +  1
)  e.  NN )  ->  ( A ^
( N  +  1 ) )  =  (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  ( N  +  1 ) ) )
2317, 22sylan2 274 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  ( N  +  1 ) ) )
24 expinnval 9423 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  (  seq 1 (  x.  , 
( NN  X.  { A } ) ,  CC ) `  N )
)
2524oveq1d 5555 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  x.  A
)  =  ( (  seq 1 (  x.  ,  ( NN  X.  { A } ) ,  CC ) `  N
)  x.  A ) )
2621, 23, 253eqtr4d 2098 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ ( N  +  1 ) )  =  ( ( A ^ N )  x.  A ) )
27 exp1 9426 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
28 mulid2 7083 . . . . . 6  |-  ( A  e.  CC  ->  (
1  x.  A )  =  A )
2927, 28eqtr4d 2091 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  ( 1  x.  A
) )
3029adantr 265 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
1 )  =  ( 1  x.  A ) )
31 simpr 107 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  N  =  0 )
3231oveq1d 5555 . . . . . 6  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  ( 0  +  1 ) )
33 0p1e1 8104 . . . . . 6  |-  ( 0  +  1 )  =  1
3432, 33syl6eq 2104 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( N  + 
1 )  =  1 )
3534oveq2d 5556 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( A ^ 1 ) )
36 oveq2 5548 . . . . . 6  |-  ( N  =  0  ->  ( A ^ N )  =  ( A ^ 0 ) )
37 exp0 9424 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3836, 37sylan9eqr 2110 . . . . 5  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^ N )  =  1 )
3938oveq1d 5555 . . . 4  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( ( A ^ N )  x.  A )  =  ( 1  x.  A ) )
4030, 35, 393eqtr4d 2098 . . 3  |-  ( ( A  e.  CC  /\  N  =  0 )  ->  ( A ^
( N  +  1 ) )  =  ( ( A ^ N
)  x.  A ) )
4126, 40jaodan 721 . 2  |-  ( ( A  e.  CC  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( A ^ ( N  + 
1 ) )  =  ( ( A ^ N )  x.  A
) )
421, 41sylan2b 275 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 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403    X. cxp 4371   ` cfv 4930  (class class class)co 5540   CCcc 6945   0cc0 6947   1c1 6948    + caddc 6950    x. cmul 6952   NNcn 7990   NN0cn0 8239   ZZ>=cuz 8569    seqcseq 9375   ^cexp 9419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-iseq 9376  df-iexp 9420
This theorem is referenced by:  expcllem  9431  expm1t  9448  expap0  9450  mulexp  9459  expadd  9462  expmul  9465  leexp2r  9474  leexp1a  9475  sqval  9478  cu2  9517  i3  9520  binom3  9534  bernneq  9537  expp1d  9550  faclbnd  9609  faclbnd2  9610  faclbnd6  9612  cjexp  9721  absexp  9906  oddpwdclemodd  10260
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