Theorem expcnv 12271

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 9566	. . . 4 |- NN C_ NN0
2		resmpt 5111	. . . 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 11947	. . . . 5 |- ( ph -> ( abs ` A ) e. RR )
6		expcnv.2	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
7	4	absge0d 11950	. . . . 5 |- ( ph -> 0 <_ ( abs ` A ) )
8	5, 6, 7	expcnvre 12270	. . . 4 |- ( ph -> ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
9		nnuz 9958	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
10	9	reseq2i 5060	. . . . . 6 |- ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) = ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) )
11	10	breq1i 4137	. . . . 5 |- ( ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` NN ) ~~> 0 <-> ( ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) |` ( ZZ>= ` 1 ) ) ~~> 0 )
12		1z 9670	. . . . . 6 |- 1 e. ZZ
13		nn0ex 9569	. . . . . . 7 |- NN0 e. _V
14	13	mptex 5943	. . . . . 6 |- ( n e. NN0 |-> ( ( abs ` A ) ^ n ) ) e. _V
15		climres 12069	. . . . . 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 430	. . . . 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 4166	. 2 |- ( ph -> ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ~~> 0 )
20		1zzd 9671	. . 3 |- ( ph -> 1 e. ZZ )
21	13	mptex 5943	. . . 4 |- ( n e. NN0 |-> ( A ^ n ) ) e. _V
22	21	a1i 9	. . 3 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) e. _V )
23		nnex 9310	. . . . 5 |- NN e. _V
24	23	mptex 5943	. . . 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 9570	. . . . . 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 11111	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( A ^ k ) e. CC )
30		oveq2 6093	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
31		eqid 2238	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
32	30, 31	fvmptg 5781	. . . . 5 |- ( ( k e. NN0 /\ ( A ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
33	27, 29, 32	syl2anc 415	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
34	33, 29	eqeltrd 2315	. . 3 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. CC )
35		absexp 11845	. . . . 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 5699	. . . 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 8354	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` A ) e. CC )
41	40, 27	expcld 11111	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( abs ` A ) ^ k ) e. CC )
42		oveq2 6093	. . . . . 6 |- ( n = k -> ( ( abs ` A ) ^ n ) = ( ( abs ` A ) ^ k ) )
43		eqid 2238	. . . . . 6 |- ( n e. NN |-> ( ( abs ` A ) ^ n ) ) = ( n e. NN |-> ( ( abs ` A ) ^ n ) )
44	42, 43	fvmptg 5781	. . . . 5 |- ( ( k e. NN /\ ( ( abs ` A ) ^ k ) e. CC ) -> ( ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
45	38, 41, 44	syl2anc 415	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( n e. NN |-> ( ( abs ` A ) ^ n ) ) ` k ) = ( ( abs ` A ) ^ k ) )
46	36, 37, 45	3eqtr4rd 2282	. . 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 12073	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4130   |-> cmpt 4192   |` cres 4776   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   < clt 8360   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ^cexp 10975   abscabs 11763   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045

This theorem is used by:  explecnv  12272  geolim  12278  geo2lim  12283