Theorem efcvgfsum 12308

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 9552	. . . . . . 7 |- ( ( A e. CC /\ n e. NN0 ) -> 0 e. ZZ )
2		nn0z 9560	. . . . . . . 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 10756	. . . . . 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 10411	. . . . . . . 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 12295	. . . . . . 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 12041	. . . . 5 |- ( ( A e. CC /\ n e. NN0 ) -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
11	10	ralrimiva 2606	. . . 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 5466	. . . 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 9852	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
16		0zd 9552	. . . . 5 |- ( A e. CC -> 0 e. ZZ )
17		eqid 2231	. . . . . . 7 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
18	17	eftvalcn 12298	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
19	18, 8	eqeltrd 2308	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
20	15, 16, 19	serf 10808	. . . 4 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) : NN0 --> CC )
21	20	ffnd 5490	. . 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 9552	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> 0 e. ZZ )
24	22	nn0zd 9661	. . . . . . 7 |- ( ( A e. CC /\ j e. NN0 ) -> j e. ZZ )
25	23, 24	fzfigd 10756	. . . . . 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 10411	. . . . . . . 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 12041	. . . . 5 |- ( ( A e. CC /\ j e. NN0 ) -> sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) e. CC )
31		oveq2 6036	. . . . . . 7 |- ( n = j -> ( 0 ... n ) = ( 0 ... j ) )
32	31	sumeq1d 12006	. . . . . 6 |- ( n = j -> sum_ k e. ( 0 ... n ) ( ( A ^ k ) / ( ! ` k ) ) = sum_ k e. ( 0 ... j ) ( ( A ^ k ) / ( ! ` k ) ) )
33	32, 12	fvmptg 5731	. . . . 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 9855	. . . . . . . 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 2324	. . . . 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 12038	. . . 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 2264	. . 3 |- ( ( A e. CC /\ j e. NN0 ) -> ( F ` j ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` j ) )
44	14, 21, 43	eqfnfvd 5756	. 2 |- ( A e. CC -> F = seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) )
45	17	efcvg 12307	. 2 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ~~> ( exp ` A ) )
46	44, 45	eqbrtrd 4115	1 |- ( A e. CC -> F ~~> ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   class class class wbr 4093   |-> cmpt 4155   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   CCcc 8090   0cc0 8092   + caddc 8095   / cdiv 8911   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305   seqcseq 10772   ^cexp 10863   !cfa 11050   ~~> cli 11918   sum_csu 11993   expce 12283

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ico 10190  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289

This theorem is referenced by: (None)