Theorem expcnv 12010

Description: A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)

Hypotheses

Ref	Expression
expcnv.1	|- ( ph -> A e. CC )
expcnv.2	|- ( ph -> ( abs ` A ) < 1 )

Assertion

Ref	Expression
expcnv	|- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Distinct variable group:   A, n

Allowed substitution hint:   ph( n)

Proof of Theorem expcnv

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnssnn0 9368	. . . 4 |- NN C_ NN0
2		resmpt 5052	. . . 4 |- ( NN C_ NN0 -> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) )
4		expcnv.1	. . . . . 6 |- ( ph -> A e. CC )
5	4	abscld 11687	. . . . 5 |- ( ph -> ( abs ` A ) e. RR )
6		expcnv.2	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
7	4	absge0d 11690	. . . . 5 |- ( ph -> 0 <_ ( abs ` A ) )
8	5, 6, 7	expcnvre 12009	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
9		nnuz 9754	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
10	9	reseq2i 5001	. . . . . 6 |- ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) )
11	10	breq1i 4089	. . . . 5 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 <-> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 )
12		1z 9468	. . . . . 6 |- 1 e. ZZ
13		nn0ex 9371	. . . . . . 7 |- NN0 e. _V
14	13	mptex 5864	. . . . . 6 |- ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) e. _V
15		climres 11809	. . . . . 6 |- ( ( 1 e. ZZ /\ ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) e. _V ) -> ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 <-> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 ) )
16	12, 14, 15	mp2an 426	. . . . 5 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 <-> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
17	11, 16	bitri 184	. . . 4 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 <-> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
18	8, 17	sylibr 134	. . 3 |- ( ph -> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 )
19	3, 18	eqbrtrrid 4118	. 2 |- ( ph -> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
20		1zzd 9469	. . 3 |- ( ph -> 1 e. ZZ )
21	13	mptex 5864	. . . 4 |- ( n e. NN0 |-> ( A ^ n ) ) e. _V
22	21	a1i 9	. . 3 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) e. _V )
23		nnex 9112	. . . . 5 |- NN e. _V
24	23	mptex 5864	. . . 4 |- ( n e. NN |-> ( ( abs ` A ) ^ n ) ) e. _V
25	24	a1i 9	. . 3 |- ( ph -> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) e. _V )
26		nnnn0 9372	. . . . . 6 |- ( k e. NN -> k e. NN0 )
27	26	adantl 277	. . . . 5 |- ( ( ph /\ k e. NN ) -> k e. NN0 )
28	4	adantr 276	. . . . . 6 |- ( ( ph /\ k e. NN ) -> A e. CC )
29	28, 27	expcld 10890	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( A ^ k ) e. CC )
30		oveq2 6008	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31		eqid 2229	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
32	30, 31	fvmptg 5709	. . . . 5 |- ( ( k e. NN0 /\ ( A ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
33	27, 29, 32	syl2anc 411	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
34	33, 29	eqeltrd 2306	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. CC )
35		absexp 11585	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) )
36	4, 26, 35	syl2an 289	. . . 4 |- ( ( ph /\ k e. NN ) -> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) )
37	33	fveq2d 5630	. . . 4 |- ( ( ph /\ k e. NN ) -> ( abs ` ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) ) = ( abs ` ( A ^ k ) ) )
38		simpr 110	. . . . 5 |- ( ( ph /\ k e. NN ) -> k e. NN )
39	5	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( abs ` A ) e. RR )
40	39	recnd 8171	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` A ) e. CC )
41	40, 27	expcld 10890	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( abs ` A ) ^ k ) e. CC )
42		oveq2 6008	. . . . . 6 |- ( n = k -> ( ( abs ` A ) ^ n ) = ( ( abs ` A ) ^ k ) )
43		eqid 2229	. . . . . 6 |- ( n e. NN |-> ( ( abs ` A ) ^ n ) ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) )
44	42, 43	fvmptg 5709	. . . . 5 |- ( ( k e. NN /\ ( ( abs ` A ) ^ k ) e. CC ) -> ( ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
45	38, 41, 44	syl2anc 411	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
46	36, 37, 45	3eqtr4rd 2273	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( abs ` ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) ) )
47	9, 20, 22, 25, 34, 46	climabs0 11813	. 2 |- ( ph -> ( ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 <-> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ~~> 0 ) )
48	19, 47	mpbird 167	1 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   class class class wbr 4082   |-> cmpt 4144   |` cres 4720   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   < clt 8177   NNcn 9106   NN0cn0 9365   ZZcz 9442   ZZ>=cuz 9718   ^cexp 10755   abscabs 11503   ~~> cli 11784

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785

This theorem is referenced by:  explecnv  12011  geolim  12017  geo2lim  12022