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Theorem expp1 10854
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 9446 . 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> N e. NN )
3 elnnuz 9837 . . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
42, 3sylib 122 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> N e. ( ZZ>= ` 1 ) )
5 simpll 527 . . . . . . 7 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
6 elnnuz 9837 . . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
7 fvconst2g 5876 . . . . . . . . . 10 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
87eleq1d 2300 . . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
96, 8sylan2br 288 . . . . . . . 8 |- ( ( A e. CC /\ x e. ( ZZ>= ` 1 ) ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
109adantlr 477 . . . . . . 7 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
115, 10mpbird 167 . . . . . 6 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
12 mulcl 8202 . . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
1312adantl 277 . . . . . 6 |- ( ( ( A e. CC /\ N e. NN ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
144, 11, 13seq3p1 10773 . . . . 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 9197 . . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
16 fvconst2g 5876 . . . . . . 7 |- ( ( A e. CC /\ ( N + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( N + 1 ) ) = A )
1715, 16sylan2 286 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( ( NN X. { A } ) ` ( N + 1 ) ) = A )
1817oveq2d 6044 . . . . 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 2264 . . . 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 10850 . . . . 5 |- ( ( A e. CC /\ ( N + 1 ) e. NN ) -> ( A ^ ( N + 1 ) ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( N + 1 ) ) )
2115, 20sylan2 286 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( N + 1 ) ) )
22 expnnval 10850 . . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
2322oveq1d 6043 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) x. A ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) x. A ) )
2419, 21, 233eqtr4d 2274 . . 3 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
25 exp1 10853 . . . . . 6 |- ( A e. CC -> ( A ^ 1 ) = A )
26 mullid 8220 . . . . . 6 |- ( A e. CC -> ( 1 x. A ) = A )
2725, 26eqtr4d 2267 . . . . 5 |- ( A e. CC -> ( A ^ 1 ) = ( 1 x. A ) )
2827adantr 276 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ 1 ) = ( 1 x. A ) )
29 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> N = 0 )
3029oveq1d 6043 . . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = ( 0 + 1 ) )
31 0p1e1 9299 . . . . . 6 |- ( 0 + 1 ) = 1
3230, 31eqtrdi 2280 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = 1 )
3332oveq2d 6044 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( A ^ 1 ) )
34 oveq2 6036 . . . . . 6 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
35 exp0 10851 . . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3634, 35sylan9eqr 2286 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
3736oveq1d 6043 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( ( A ^ N ) x. A ) = ( 1 x. A ) )
3828, 33, 373eqtr4d 2274 . . 3 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
3924, 38jaodan 805 . 2 |- ( ( A e. CC /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
401, 39sylan2b 287 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 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   {csn 3673   X. cxp 4729   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   NNcn 9185   NN0cn0 9444   ZZ>=cuz 9799   seqcseq 10755   ^cexp 10846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-seqfrec 10756  df-exp 10847
This theorem is referenced by:  expcllem  10858  expm1t  10875  expap0  10877  mulexp  10886  expadd  10889  expmul  10892  leexp2r  10901  leexp1a  10902  sqval  10905  cu2  10946  i3  10949  binom3  10965  bernneq  10968  modqexp  10974  expp1d  10982  faclbnd  11049  faclbnd2  11050  faclbnd6  11052  cjexp  11516  absexp  11702  binomlem  12107  geolim  12135  geo2sum  12138  efexp  12306  demoivreALT  12398  prmdvdsexp  12783  oddpwdclemodd  12807  pcexp  12945  numexpp1  13060  2exp7  13070  cnfldexp  14656  expcn  15363  expcncf  15403  dvexp  15505  tangtx  15632  rpcxpmul2  15707  binom4  15773  perfectlem1  15796  perfectlem2  15797
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