ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expp1 Unicode version
Theorem expp1 10458
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 9112 . 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2 simpr 109 . . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> N e. NN )
3 elnnuz 9498 . . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
42, 3sylib 121 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> N e. ( ZZ>= ` 1 ) )
5 simpll 519 . . . . . . 7 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
6 elnnuz 9498 . . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
7 fvconst2g 5698 . . . . . . . . . 10 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
87eleq1d 2234 . . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
96, 8sylan2br 286 . . . . . . . 8 |- ( ( A e. CC /\ x e. ( ZZ>= ` 1 ) ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
109adantlr 469 . . . . . . 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 7876 . . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
1312adantl 275 . . . . . 6 |- ( ( ( A e. CC /\ N e. NN ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
144, 11, 13seq3p1 10393 . . . . 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 8865 . . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
16 fvconst2g 5698 . . . . . . 7 |- ( ( A e. CC /\ ( N + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( N + 1 ) ) = A )
1715, 16sylan2 284 . . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( ( NN X. { A } ) ` ( N + 1 ) ) = A )
1817oveq2d 5857 . . . . 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 2198 . . . 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 10454 . . . . 5 |- ( ( A e. CC /\ ( N + 1 ) e. NN ) -> ( A ^ ( N + 1 ) ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( N + 1 ) ) )
2115, 20sylan2 284 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` ( N + 1 ) ) )
22 expnnval 10454 . . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) )
2322oveq1d 5856 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) x. A ) = ( ( seq 1 ( x. , ( NN X. { A } ) ) ` N ) x. A ) )
2419, 21, 233eqtr4d 2208 . . 3 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
25 exp1 10457 . . . . . 6 |- ( A e. CC -> ( A ^ 1 ) = A )
26 mulid2 7893 . . . . . 6 |- ( A e. CC -> ( 1 x. A ) = A )
2725, 26eqtr4d 2201 . . . . 5 |- ( A e. CC -> ( A ^ 1 ) = ( 1 x. A ) )
2827adantr 274 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ 1 ) = ( 1 x. A ) )
29 simpr 109 . . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> N = 0 )
3029oveq1d 5856 . . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = ( 0 + 1 ) )
31 0p1e1 8967 . . . . . 6 |- ( 0 + 1 ) = 1
3230, 31eqtrdi 2214 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = 1 )
3332oveq2d 5857 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( A ^ 1 ) )
34 oveq2 5849 . . . . . 6 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
35 exp0 10455 . . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
3634, 35sylan9eqr 2220 . . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
3736oveq1d 5856 . . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( ( A ^ N ) x. A ) = ( 1 x. A ) )
3828, 33, 373eqtr4d 2208 . . 3 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
3924, 38jaodan 787 . 2 |- ( ( A e. CC /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
401, 39sylan2b 285 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 698   = wceq 1343   e. wcel 2136   {csn 3575   X. cxp 4601   ` cfv 5187  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750   + caddc 7752   x. cmul 7754   NNcn 8853   NN0cn0 9110   ZZ>=cuz 9462   seqcseq 10376   ^cexp 10450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-n0 9111  df-z 9188  df-uz 9463  df-seqfrec 10377  df-exp 10451
This theorem is referenced by:  expcllem  10462  expm1t  10479  expap0  10481  mulexp  10490  expadd  10493  expmul  10496  leexp2r  10505  leexp1a  10506  sqval  10509  cu2  10549  i3  10552  binom3  10568  bernneq  10571  modqexp  10577  expp1d  10585  faclbnd  10650  faclbnd2  10651  faclbnd6  10653  cjexp  10831  absexp  11017  binomlem  11420  geolim  11448  geo2sum  11451  efexp  11619  demoivreALT  11710  prmdvdsexp  12076  oddpwdclemodd  12100  pcexp  12237  expcncf  13192  dvexp  13275  tangtx  13359  binom4  13497
  Copyright terms: Public domain W3C validator