Theorem expp1 9483

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.

Step	Hyp	Ref	Expression
1		elnn0 8290	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		simpr 108	. . . . . . 7 |- ( ( A e. CC /\ N e. NN ) -> N e. NN )
3		elnnuz 8655	. . . . . . 7 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
4	2, 3	sylib 120	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> N e. ( ZZ>= ` 1 ) )
5		cnex 7097	. . . . . . 7 |- CC e. _V
6	5	a1i 9	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> CC e. _V )
7		simpll 495	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> A e. CC )
8		elnnuz 8655	. . . . . . . . 9 |- ( x e. NN <-> x e. ( ZZ>= ` 1 ) )
9		fvconst2g 5396	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. NN ) -> ( ( NN X. { A } ) ` x ) = A )
10	9	eleq1d 2147	. . . . . . . . 9 |- ( ( A e. CC /\ x e. NN ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
11	8, 10	sylan2br 282	. . . . . . . 8 |- ( ( A e. CC /\ x e. ( ZZ>= ` 1 ) ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
12	11	adantlr 460	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( ( NN X. { A } ) ` x ) e. CC <-> A e. CC ) )
13	7, 12	mpbird 165	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { A } ) ` x ) e. CC )
14		mulcl 7100	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
15	14	adantl 271	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
16	4, 6, 13, 15	iseqp1 9445	. . . . 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 8051	. . . . . . 7 |- ( N e. NN -> ( N + 1 ) e. NN )
18		fvconst2g 5396	. . . . . . 7 |- ( ( A e. CC /\ ( N + 1 ) e. NN ) -> ( ( NN X. { A } ) ` ( N + 1 ) ) = A )
19	17, 18	sylan2 280	. . . . . 6 |- ( ( A e. CC /\ N e. NN ) -> ( ( NN X. { A } ) ` ( N + 1 ) ) = A )
20	19	oveq2d 5548	. . . . 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 ) )
21	16, 20	eqtrd 2113	. . . 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 9479	. . . . 5 |- ( ( A e. CC /\ ( N + 1 ) e. NN ) -> ( A ^ ( N + 1 ) ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( N + 1 ) ) )
23	17, 22	sylan2 280	. . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` ( N + 1 ) ) )
24		expinnval 9479	. . . . 5 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) )
25	24	oveq1d 5547	. . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) x. A ) = ( ( seq 1 ( x. , ( NN X. { A } ) , CC ) ` N ) x. A ) )
26	21, 23, 25	3eqtr4d 2123	. . 3 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
27		exp1 9482	. . . . . 6 |- ( A e. CC -> ( A ^ 1 ) = A )
28		mulid2 7117	. . . . . 6 |- ( A e. CC -> ( 1 x. A ) = A )
29	27, 28	eqtr4d 2116	. . . . 5 |- ( A e. CC -> ( A ^ 1 ) = ( 1 x. A ) )
30	29	adantr 270	. . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ 1 ) = ( 1 x. A ) )
31		simpr 108	. . . . . . 7 |- ( ( A e. CC /\ N = 0 ) -> N = 0 )
32	31	oveq1d 5547	. . . . . 6 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = ( 0 + 1 ) )
33		0p1e1 8153	. . . . . 6 |- ( 0 + 1 ) = 1
34	32, 33	syl6eq 2129	. . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( N + 1 ) = 1 )
35	34	oveq2d 5548	. . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( A ^ 1 ) )
36		oveq2 5540	. . . . . 6 |- ( N = 0 -> ( A ^ N ) = ( A ^ 0 ) )
37		exp0 9480	. . . . . 6 |- ( A e. CC -> ( A ^ 0 ) = 1 )
38	36, 37	sylan9eqr 2135	. . . . 5 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ N ) = 1 )
39	38	oveq1d 5547	. . . 4 |- ( ( A e. CC /\ N = 0 ) -> ( ( A ^ N ) x. A ) = ( 1 x. A ) )
40	30, 35, 39	3eqtr4d 2123	. . 3 |- ( ( A e. CC /\ N = 0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
41	26, 40	jaodan 743	. 2 |- ( ( A e. CC /\ ( N e. NN \/ N = 0 ) ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )
42	1, 41	sylan2b 281	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   X. cxp 4361   ` cfv 4922  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   NNcn 8039   NN0cn0 8288   ZZ>=cuz 8619   seqcseq 9431   ^cexp 9475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476

This theorem is referenced by:  expcllem  9487  expm1t  9504  expap0  9506  mulexp  9515  expadd  9518  expmul  9521  leexp2r  9530  leexp1a  9531  sqval  9534  cu2  9573  i3  9576  binom3  9590  bernneq  9593  expp1d  9606  faclbnd  9668  faclbnd2  9669  faclbnd6  9671  cjexp  9780  absexp  9965  prmdvdsexp  10527  oddpwdclemodd  10550