Theorem efcvgfsum 11716

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 9300	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> 0 e. ZZ )
2		nn0z 9308	. . . . . . . 8 |- ( n e. NN0 -> n e. ZZ )
3	2	adantl 277	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> n e. ZZ )
4	1, 3	fzfigd 10468	. . . . . 6 |- ( ( A e. CC /\ n e. NN0 ) -> ( 0 ... n ) e. Fin )
5		simpll 527	. . . . . . 7 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> A e. CC )
6		elfznn0 10150	. . . . . . . 8 |- ( k e. ( 0 ... n ) -> k e. NN0 )
7	6	adantl 277	. . . . . . 7 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> k e. NN0 )
8		eftcl 11703	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
9	5, 7, 8	syl2anc 411	. . . . . 6 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
10	4, 9	fsumcl 11449	. . . . 5 |- ( ( A e. CC /\ n e. NN0 ) -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
11	10	ralrimiva 2563	. . . 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 5364	. . . 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 9598	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
16		0zd 9300	. . . . 5 |- ( A e. CC -> 0 e. ZZ )
17		eqid 2189	. . . . . . 7 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
18	17	eftvalcn 11706	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
19	18, 8	eqeltrd 2266	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
20	15, 16, 19	serf 10513	. . . 4 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) : NN0 --> CC )
21	20	ffnd 5388	. . 3 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) Fn NN0 )
22		simpr 110	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> j e. NN0 )
23		0zd 9300	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> 0 e. ZZ )
24	22	nn0zd 9408	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ZZ )
25	23, 24	fzfigd 10468	. . . . . 6 |- ( ( A e. CC /\ j e. NN0 ) -> ( 0 ... j ) e. Fin )
26		simpll 527	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> A e. CC )
27		elfznn0 10150	. . . . . . . 8 |- ( k e. ( 0 ... j ) -> k e. NN0 )
28	27	adantl 277	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> k e. NN0 )
29	26, 28, 8	syl2anc 411	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
30	25, 29	fsumcl 11449	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
31		oveq2 5908	. . . . . . 7 |- ( n = j -> ( 0 ... n ) = ( 0 ... j ) )
32	31	sumeq1d 11415	. . . . . 6 |- ( n = j -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
33	32, 12	fvmptg 5616	. . . . 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 411	. . . 4 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
35		simpll 527	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> A e. CC )
36		elnn0uz 9601	. . . . . . . 8 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
37	36	biimpri 133	. . . . . . 7 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
38	37	adantl 277	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
39	35, 38, 18	syl2anc 411	. . . . 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 2282	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ( ZZ>= ` 0 ) )
41	35, 38, 8	syl2anc 411	. . . . 5 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
42	39, 40, 41	fsum3ser 11446	. . . 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 2222	. . 3 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` j ) )
44	14, 21, 43	eqfnfvd 5640	. 2 |- ( A e. CC -> F = seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) )
45	17	efcvg 11715	. 2 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ~~> ( exp ` A ) )
46	44, 45	eqbrtrd 4043	1 |- ( A e. CC -> F ~~> ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   class class class wbr 4021   |-> cmpt 4082   Fn wfn 5233   ` cfv 5238  (class class class)co 5900   CCcc 7844   0cc0 7846   + caddc 7849   / cdiv 8664   NN0cn0 9211   ZZcz 9288   ZZ>=cuz 9563   ...cfz 10044   seqcseq 10484   ^cexp 10559   !cfa 10746   ~~> cli 11327   sum_csu 11402   expce 11691

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 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966

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 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-isom 5247  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-frec 6420  df-1o 6445  df-oadd 6449  df-er 6563  df-en 6771  df-dom 6772  df-fin 6773  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-n0 9212  df-z 9289  df-uz 9564  df-q 9656  df-rp 9690  df-ico 9930  df-fz 10045  df-fzo 10179  df-seqfrec 10485  df-exp 10560  df-fac 10747  df-ihash 10797  df-cj 10892  df-re 10893  df-im 10894  df-rsqrt 11048  df-abs 11049  df-clim 11328  df-sumdc 11403  df-ef 11697

This theorem is referenced by: (None)