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Theorem expp1 10573
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 9219 . 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> N e. NN )
3 elnnuz 9606 . . . . . . 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 9606 . . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
7 fvconst2g 5758 . . . . . . . . . 10 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
87eleq1d 2258 . . . . . . . . 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 7978 . . . . . . 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 10507 . . . . 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 8972 . . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
16 fvconst2g 5758 . . . . . . 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 5920 . . . . 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 2222 . . . 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 10569 . . . . 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 10569 . . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
2322oveq1d 5919 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) x. A ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) x. A ) )
2419, 21, 233eqtr4d 2232 . . 3 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
25 exp1 10572 . . . . . 6 |- ( A e. CC -> ( A ^ 1 ) = A )
26 mullid 7996 . . . . . 6 |- ( A e. CC -> ( 1 x. A ) = A )
2725, 26eqtr4d 2225 . . . . 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 5919 . . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = ( 0 + 1 ) )
31 0p1e1 9074 . . . . . 6 |- ( 0 + 1 ) = 1
3230, 31eqtrdi 2238 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = 1 )
3332oveq2d 5920 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( A ^ 1 ) )
34 oveq2 5912 . . . . . 6 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
35 exp0 10570 . . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3634, 35sylan9eqr 2244 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
3736oveq1d 5919 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( ( A ^ N ) x. A ) = ( 1 x. A ) )
3828, 33, 373eqtr4d 2232 . . 3 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
3924, 38jaodan 798 . 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 709   = wceq 1364   e. wcel 2160   {csn 3614   X. cxp 4649   ` cfv 5242  (class class class)co 5904   CCcc 7849   0cc0 7851   1c1 7852   + caddc 7854   x. cmul 7856   NNcn 8960   NN0cn0 9217   ZZ>=cuz 9569   seqcseq 10490   ^cexp 10565
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612  ax-cnex 7942  ax-resscn 7943  ax-1cn 7944  ax-1re 7945  ax-icn 7946  ax-addcl 7947  ax-addrcl 7948  ax-mulcl 7949  ax-mulrcl 7950  ax-addcom 7951  ax-mulcom 7952  ax-addass 7953  ax-mulass 7954  ax-distr 7955  ax-i2m1 7956  ax-0lt1 7957  ax-1rid 7958  ax-0id 7959  ax-rnegex 7960  ax-precex 7961  ax-cnre 7962  ax-pre-ltirr 7963  ax-pre-ltwlin 7964  ax-pre-lttrn 7965  ax-pre-apti 7966  ax-pre-ltadd 7967  ax-pre-mulgt0 7968  ax-pre-mulext 7969
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-ilim 4394  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-riota 5859  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-recs 6338  df-frec 6424  df-pnf 8035  df-mnf 8036  df-xr 8037  df-ltxr 8038  df-le 8039  df-sub 8171  df-neg 8172  df-reap 8573  df-ap 8580  df-div 8671  df-inn 8961  df-n0 9218  df-z 9295  df-uz 9570  df-seqfrec 10491  df-exp 10566
This theorem is referenced by:  expcllem  10577  expm1t  10594  expap0  10596  mulexp  10605  expadd  10608  expmul  10611  leexp2r  10620  leexp1a  10621  sqval  10624  cu2  10665  i3  10668  binom3  10684  bernneq  10687  modqexp  10693  expp1d  10701  faclbnd  10768  faclbnd2  10769  faclbnd6  10771  cjexp  10949  absexp  11135  binomlem  11538  geolim  11566  geo2sum  11569  efexp  11737  demoivreALT  11828  prmdvdsexp  12197  oddpwdclemodd  12221  pcexp  12358  cnfldexp  13926  expcncf  14616  dvexp  14713  tangtx  14797  binom4  14935
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