Theorem efcvgfsum 11813

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 9332	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> 0 e. ZZ )
2		nn0z 9340	. . . . . . . 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 10505	. . . . . 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 10183	. . . . . . . 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 11800	. . . . . . 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 11546	. . . . 5 |- ( ( A e. CC /\ n e. NN0 ) -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
11	10	ralrimiva 2567	. . . 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 5381	. . . 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 9630	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
16		0zd 9332	. . . . 5 |- ( A e. CC -> 0 e. ZZ )
17		eqid 2193	. . . . . . 7 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
18	17	eftvalcn 11803	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
19	18, 8	eqeltrd 2270	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
20	15, 16, 19	serf 10557	. . . 4 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) : NN0 --> CC )
21	20	ffnd 5405	. . 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 9332	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> 0 e. ZZ )
24	22	nn0zd 9440	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ZZ )
25	23, 24	fzfigd 10505	. . . . . 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 10183	. . . . . . . 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 11546	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
31		oveq2 5927	. . . . . . 7 |- ( n = j -> ( 0 ... n ) = ( 0 ... j ) )
32	31	sumeq1d 11512	. . . . . 6 |- ( n = j -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
33	32, 12	fvmptg 5634	. . . . 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 9633	. . . . . . . 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 2286	. . . . 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 11543	. . . 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 2226	. . 3 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` j ) )
44	14, 21, 43	eqfnfvd 5659	. 2 |- ( A e. CC -> F = seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) )
45	17	efcvg 11812	. 2 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ~~> ( exp ` A ) )
46	44, 45	eqbrtrd 4052	1 |- ( A e. CC -> F ~~> ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   class class class wbr 4030   |-> cmpt 4091   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   + caddc 7877   / cdiv 8693   NN0cn0 9243   ZZcz 9320   ZZ>=cuz 9595   ...cfz 10077   seqcseq 10521   ^cexp 10612   !cfa 10799   ~~> cli 11424   sum_csu 11499   expce 11788

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ico 9963  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ef 11794

This theorem is referenced by: (None)