ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expp1 Unicode version
Theorem expp1 10996
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 9569 . 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> N e. NN )
3 elnnuz 9968 . . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
42, 3sylib 122 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> N e. ( ZZ>= ` 1 ) )
5 simpll 531 . . . . . . 7 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
6 elnnuz 9968 . . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
7 fvconst2g 5929 . . . . . . . . . 10 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
87eleq1d 2307 . . . . . . . . 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 481 . . . . . . 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 8306 . . . . . . 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 10915 . . . . 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 9318 . . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
16 fvconst2g 5929 . . . . . . 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 6101 . . . . 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 2271 . . . 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 10992 . . . . 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 10992 . . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
2322oveq1d 6100 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) x. A ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) x. A ) )
2419, 21, 233eqtr4d 2281 . . 3 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
25 exp1 10995 . . . . . 6 |- ( A e. CC -> ( A ^ 1 ) = A )
26 mullid 8324 . . . . . 6 |- ( A e. CC -> ( 1 x. A ) = A )
2725, 26eqtr4d 2274 . . . . 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 6100 . . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = ( 0 + 1 ) )
31 0p1e1 9420 . . . . . 6 |- ( 0 + 1 ) = 1
3230, 31eqtrdi 2287 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = 1 )
3332oveq2d 6101 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( A ^ 1 ) )
34 oveq2 6093 . . . . . 6 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
35 exp0 10993 . . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3634, 35sylan9eqr 2293 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
3736oveq1d 6100 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( ( A ^ N ) x. A ) = ( 1 x. A ) )
3828, 33, 373eqtr4d 2281 . . 3 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
3924, 38jaodan 809 . 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
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   {csn 3709   X. cxp 4772   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NNcn 9306   NN0cn0 9567   ZZ>=cuz 9930   seqcseq 10897   ^cexp 10988
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-n0 9568  df-z 9649  df-uz 9931  df-seqfrec 10898  df-exp 10989
This theorem is used by:  expcllem  11000  expm1t  11017  expap0  11019  mulexp  11028  expadd  11031  expmul  11034  leexp2r  11043  leexp1a  11044  sqval  11047  cu2  11088  i3  11091  binom3  11107  bernneq  11111  modqexp  11117  expp1d  11125  faclbnd  11193  faclbnd2  11194  faclbnd6  11196  cjexp  11672  absexp  11860  binomlem  12266  geolim  12294  geo2sum  12297  efexp  12465  demoivreALT  12557  prmdvdsexp  12943  pcexp  13108  numexpp1  13224  2exp7  13234  cnfldexp  14963  expcn  15719  expcncf  15759  dvexp  15861  tangtx  15989  rpcxpmul2  16068  binom4  16138  log2ublem2  16141  log2ublog2  16143  perfectlem1  16197  perfectlem2  16198  bclbnd  16205
  Copyright terms: Public domain W3C validator