Theorem efcvgfsum 11630

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 9224	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> 0 e. ZZ )
2		nn0z 9232	. . . . . . . 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 10387	. . . . . 6 |- ( ( A e. CC /\ n e. NN0 ) -> ( 0 ... n ) e. Fin )
5		simpll 524	. . . . . . 7 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> A e. CC )
6		elfznn0 10070	. . . . . . . 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 11617	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
9	5, 7, 8	syl2anc 409	. . . . . 6 |- ( ( ( A e. CC /\ n e. NN0 ) /\ k e. ( 0 ... n ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
10	4, 9	fsumcl 11363	. . . . 5 |- ( ( A e. CC /\ n e. NN0 ) -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
11	10	ralrimiva 2543	. . . 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 5324	. . . 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 9521	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
16		0zd 9224	. . . . 5 |- ( A e. CC -> 0 e. ZZ )
17		eqid 2170	. . . . . . 7 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
18	17	eftvalcn 11620	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
19	18, 8	eqeltrd 2247	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
20	15, 16, 19	serf 10430	. . . 4 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) : NN0 --> CC )
21	20	ffnd 5348	. . 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 9224	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> 0 e. ZZ )
24	22	nn0zd 9332	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ZZ )
25	23, 24	fzfigd 10387	. . . . . 6 |- ( ( A e. CC /\ j e. NN0 ) -> ( 0 ... j ) e. Fin )
26		simpll 524	. . . . . . 7 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> A e. CC )
27		elfznn0 10070	. . . . . . . 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 409	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( 0 ... j ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
30	25, 29	fsumcl 11363	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
31		oveq2 5861	. . . . . . 7 |- ( n = j -> ( 0 ... n ) = ( 0 ... j ) )
32	31	sumeq1d 11329	. . . . . 6 |- ( n = j -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
33	32, 12	fvmptg 5572	. . . . 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 409	. . . 4 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
35		simpll 524	. . . . . 6 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> A e. CC )
36		elnn0uz 9524	. . . . . . . 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 409	. . . . 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 2263	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ( ZZ>= ` 0 ) )
41	35, 38, 8	syl2anc 409	. . . . 5 |- ( ( ( A e. CC /\ j e. NN0 ) /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
42	39, 40, 41	fsum3ser 11360	. . . 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 2203	. . 3 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` j ) )
44	14, 21, 43	eqfnfvd 5596	. 2 |- ( A e. CC -> F = seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) )
45	17	efcvg 11629	. 2 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ~~> ( exp ` A ) )
46	44, 45	eqbrtrd 4011	1 |- ( A e. CC -> F ~~> ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   class class class wbr 3989   |-> cmpt 4050   Fn wfn 5193   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   + caddc 7777   / cdiv 8589   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401   ^cexp 10475   !cfa 10659   ~~> cli 11241   sum_csu 11316   expce 11605

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611

This theorem is referenced by: (None)