Theorem expcnv 11547

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 9210	. . . 4 |- NN C_ NN0
2		resmpt 4973	. . . 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 11225	. . . . 5 |- ( ph -> ( abs ` A ) e. RR )
6		expcnv.2	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
7	4	absge0d 11228	. . . . 5 |- ( ph -> 0 <_ ( abs ` A ) )
8	5, 6, 7	expcnvre 11546	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
9		nnuz 9595	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
10	9	reseq2i 4922	. . . . . 6 |- ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) )
11	10	breq1i 4025	. . . . 5 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 <-> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 )
12		1z 9310	. . . . . 6 |- 1 e. ZZ
13		nn0ex 9213	. . . . . . 7 |- NN0 e. _V
14	13	mptex 5763	. . . . . 6 |- ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) e. _V
15		climres 11346	. . . . . 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 4054	. 2 |- ( ph -> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
20		1zzd 9311	. . 3 |- ( ph -> 1 e. ZZ )
21	13	mptex 5763	. . . 4 |- ( n e. NN0 |-> ( A ^ n ) ) e. _V
22	21	a1i 9	. . 3 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) e. _V )
23		nnex 8956	. . . . 5 |- NN e. _V
24	23	mptex 5763	. . . 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 9214	. . . . . 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 10688	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( A ^ k ) e. CC )
30		oveq2 5905	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31		eqid 2189	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
32	30, 31	fvmptg 5613	. . . . 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 2266	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. CC )
35		absexp 11123	. . . . 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 5538	. . . 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 8017	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` A ) e. CC )
41	40, 27	expcld 10688	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( abs ` A ) ^ k ) e. CC )
42		oveq2 5905	. . . . . 6 |- ( n = k -> ( ( abs ` A ) ^ n ) = ( ( abs ` A ) ^ k ) )
43		eqid 2189	. . . . . 6 |- ( n e. NN |-> ( ( abs ` A ) ^ n ) ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) )
44	42, 43	fvmptg 5613	. . . . 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 2233	. . 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 11350	. 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 1364   e. wcel 2160   _Vcvv 2752   C_ wss 3144   class class class wbr 4018   |-> cmpt 4079   |` cres 4646   ` cfv 5235  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842   1c1 7843   < clt 8023   NNcn 8950   NN0cn0 9207   ZZcz 9284   ZZ>=cuz 9559   ^cexp 10553   abscabs 11041   ~~> cli 11321

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 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

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 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322

This theorem is referenced by:  explecnv  11548  geolim  11554  geo2lim  11559