Theorem expcnv 11669

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 9252	. . . 4 |- NN C_ NN0
2		resmpt 4994	. . . 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 11346	. . . . 5 |- ( ph -> ( abs ` A ) e. RR )
6		expcnv.2	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
7	4	absge0d 11349	. . . . 5 |- ( ph -> 0 <_ ( abs ` A ) )
8	5, 6, 7	expcnvre 11668	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
9		nnuz 9637	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
10	9	reseq2i 4943	. . . . . 6 |- ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) )
11	10	breq1i 4040	. . . . 5 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 <-> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 )
12		1z 9352	. . . . . 6 |- 1 e. ZZ
13		nn0ex 9255	. . . . . . 7 |- NN0 e. _V
14	13	mptex 5788	. . . . . 6 |- ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) e. _V
15		climres 11468	. . . . . 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 4069	. 2 |- ( ph -> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
20		1zzd 9353	. . 3 |- ( ph -> 1 e. ZZ )
21	13	mptex 5788	. . . 4 |- ( n e. NN0 |-> ( A ^ n ) ) e. _V
22	21	a1i 9	. . 3 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) e. _V )
23		nnex 8996	. . . . 5 |- NN e. _V
24	23	mptex 5788	. . . 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 9256	. . . . . 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 10765	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( A ^ k ) e. CC )
30		oveq2 5930	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31		eqid 2196	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
32	30, 31	fvmptg 5637	. . . . 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 2273	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. CC )
35		absexp 11244	. . . . 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 5562	. . . 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 8055	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` A ) e. CC )
41	40, 27	expcld 10765	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( abs ` A ) ^ k ) e. CC )
42		oveq2 5930	. . . . . 6 |- ( n = k -> ( ( abs ` A ) ^ n ) = ( ( abs ` A ) ^ k ) )
43		eqid 2196	. . . . . 6 |- ( n e. NN |-> ( ( abs ` A ) ^ n ) ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) )
44	42, 43	fvmptg 5637	. . . . 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 2240	. . 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 11472	. 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 2167   _Vcvv 2763   C_ wss 3157   class class class wbr 4033   |-> cmpt 4094   |` cres 4665   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   < clt 8061   NNcn 8990   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ^cexp 10630   abscabs 11162   ~~> cli 11443

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444

This theorem is referenced by:  explecnv  11670  geolim  11676  geo2lim  11681