Theorem efcvgfsum 11373

Description: Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Hypothesis

Ref	Expression
efcvgfsum.1	|- F = ( n e. NN0 |-> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) )

Assertion

Ref	Expression
efcvgfsum	|- ( A e. CC -> F ~~> ( exp ` A ) )

Distinct variable group:   k, n, A

Allowed substitution hints:   F( k, n)

Proof of Theorem efcvgfsum

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0zd 9066	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> 0 e. ZZ )
2		nn0z 9074	. . . . . . . 8 |- ( n e. NN0 -> n e. ZZ )
3	2	adantl 275	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> n e. ZZ )
4	1, 3	fzfigd 10204	. . . . . 6 |- ( ( A e. CC /\ n e. NN0 ) -> ( 0 ... n ) e. Fin )
5		simpll 518	. . . . . . 7 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> A e. CC )
6		elfznn0 9894	. . . . . . . 8 |- ( k e. ( 0 ... n ) -> k e. NN0 )
7	6	adantl 275	. . . . . . 7 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> k e. NN0 )
8		eftcl 11360	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
9	5, 7, 8	syl2anc 408	. . . . . 6 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
10	4, 9	fsumcl 11169	. . . . 5 |- ( ( A e. CC /\ n e. NN0 ) -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
11	10	ralrimiva 2505	. . . 4 |- ( A e. CC -> A. n e. NN0 sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
12		efcvgfsum.1	. . . . 5 |- F = ( n e. NN0 |-> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) )
13	12	fnmpt 5249	. . . 4 |- ( A. n e. NN0 sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC -> F Fn NN0 )
14	11, 13	syl 14	. . 3 |- ( A e. CC -> F Fn NN0 )
15		nn0uz 9360	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
16		0zd 9066	. . . . 5 |- ( A e. CC -> 0 e. ZZ )
17		eqid 2139	. . . . . . 7 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
18	17	eftvalcn 11363	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
19	18, 8	eqeltrd 2216	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
20	15, 16, 19	serf 10247	. . . 4 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) : NN0 --> CC )
21	20	ffnd 5273	. . 3 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) Fn NN0 )
22		simpr 109	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> j e. NN0 )
23		0zd 9066	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> 0 e. ZZ )
24	22	nn0zd 9171	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ZZ )
25	23, 24	fzfigd 10204	. . . . . 6 |- ( ( A e. CC /\ j e. NN0 ) -> ( 0 ... j ) e. Fin )
26		simpll 518	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> A e. CC )
27		elfznn0 9894	. . . . . . . 8 |- ( k e. ( 0 ... j ) -> k e. NN0 )
28	27	adantl 275	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> k e. NN0 )
29	26, 28, 8	syl2anc 408	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
30	25, 29	fsumcl 11169	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
31		oveq2 5782	. . . . . . 7 |- ( n = j -> ( 0 ... n ) = ( 0 ... j ) )
32	31	sumeq1d 11135	. . . . . 6 |- ( n = j -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
33	32, 12	fvmptg 5497	. . . . 5 |- ( ( j e. NN0 /\ sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) e. CC ) -> ( F ` j ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
34	22, 30, 33	syl2anc 408	. . . 4 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
35		simpll 518	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> A e. CC )
36		elnn0uz 9363	. . . . . . . 8 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
37	36	biimpri 132	. . . . . . 7 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
38	37	adantl 275	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
39	35, 38, 18	syl2anc 408	. . . . 5 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
40	22, 15	eleqtrdi 2232	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ( ZZ>= ` 0 ) )
41	35, 38, 8	syl2anc 408	. . . . 5 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
42	39, 40, 41	fsum3ser 11166	. . . 4 |- ( ( A e. CC /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` j ) )
43	34, 42	eqtrd 2172	. . 3 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` j ) )
44	14, 21, 43	eqfnfvd 5521	. 2 |- ( A e. CC -> F = seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) )
45	17	efcvg 11372	. 2 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ~~> ( exp ` A ) )
46	44, 45	eqbrtrd 3950	1 |- ( A e. CC -> F ~~> ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2416   class class class wbr 3929   |-> cmpt 3989   Fn wfn 5118   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   + caddc 7623   / cdiv 8432   NN0cn0 8977   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790   seqcseq 10218   ^cexp 10292   !cfa 10471   ~~> cli 11047   sum_csu 11122   expce 11348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354

This theorem is referenced by: (None)