Theorem expcnv 12215

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 9516	. . . 4 |- NN C_ NN0
2		resmpt 5091	. . . 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 11891	. . . . 5 |- ( ph -> ( abs ` A ) e. RR )
6		expcnv.2	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
7	4	absge0d 11894	. . . . 5 |- ( ph -> 0 <_ ( abs ` A ) )
8	5, 6, 7	expcnvre 12214	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
9		nnuz 9908	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
10	9	reseq2i 5040	. . . . . 6 |- ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) )
11	10	breq1i 4121	. . . . 5 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 <-> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 )
12		1z 9620	. . . . . 6 |- 1 e. ZZ
13		nn0ex 9519	. . . . . . 7 |- NN0 e. _V
14	13	mptex 5917	. . . . . 6 |- ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) e. _V
15		climres 12013	. . . . . 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 4150	. 2 |- ( ph -> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
20		1zzd 9621	. . 3 |- ( ph -> 1 e. ZZ )
21	13	mptex 5917	. . . 4 |- ( n e. NN0 |-> ( A ^ n ) ) e. _V
22	21	a1i 9	. . 3 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) e. _V )
23		nnex 9260	. . . . 5 |- NN e. _V
24	23	mptex 5917	. . . 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 9520	. . . . . 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 11060	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( A ^ k ) e. CC )
30		oveq2 6066	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31		eqid 2234	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
32	30, 31	fvmptg 5758	. . . . 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 2311	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. CC )
35		absexp 11789	. . . . 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 5679	. . . 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 8318	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` A ) e. CC )
41	40, 27	expcld 11060	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( abs ` A ) ^ k ) e. CC )
42		oveq2 6066	. . . . . 6 |- ( n = k -> ( ( abs ` A ) ^ n ) = ( ( abs ` A ) ^ k ) )
43		eqid 2234	. . . . . 6 |- ( n e. NN |-> ( ( abs ` A ) ^ n ) ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) )
44	42, 43	fvmptg 5758	. . . . 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 2278	. . 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 12017	. 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 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   |` cres 4756   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   < clt 8324   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ^cexp 10924   abscabs 11707   ~~> cli 11988

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989

This theorem is referenced by:  explecnv  12216  geolim  12222  geo2lim  12227