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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.
StepHypRef Expression
1 0zd 9300 . . . . . . 7  |-  ( ( A  e.  CC  /\  n  e.  NN0 )  -> 
0  e.  ZZ )
2 nn0z 9308 . . . . . . . 8  |-  ( n  e.  NN0  ->  n  e.  ZZ )
32adantl 277 . . . . . . 7  |-  ( ( A  e.  CC  /\  n  e.  NN0 )  ->  n  e.  ZZ )
41, 3fzfigd 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 )
76adantl 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 )
95, 7, 8syl2anc 411 . . . . . 6  |-  ( ( ( A  e.  CC  /\  n  e.  NN0 )  /\  k  e.  (
0 ... n ) )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
104, 9fsumcl 11449 . . . . 5  |-  ( ( A  e.  CC  /\  n  e.  NN0 )  ->  sum_ k  e.  ( 0 ... n ) ( ( A ^ k
)  /  ( ! `
 k ) )  e.  CC )
1110ralrimiva 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 ) ) )
1312fnmpt 5364 . . . 4  |-  ( A. n  e.  NN0  sum_ k  e.  ( 0 ... n
) ( ( A ^ k )  / 
( ! `  k
) )  e.  CC  ->  F  Fn  NN0 )
1411, 13syl 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
) ) )
1817eftvalcn 11706 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
1918, 8eqeltrd 2266 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
2015, 16, 19serf 10513 . . . 4  |-  ( A  e.  CC  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) : NN0 --> CC )
2120ffnd 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 )
2422nn0zd 9408 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ZZ )
2523, 24fzfigd 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 )
2827adantl 277 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  k  e.  NN0 )
2926, 28, 8syl2anc 411 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
3025, 29fsumcl 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
) )
3231sumeq1d 11415 . . . . . 6  |-  ( n  =  j  ->  sum_ k  e.  ( 0 ... n
) ( ( A ^ k )  / 
( ! `  k
) )  =  sum_ k  e.  ( 0 ... j ) ( ( A ^ k
)  /  ( ! `
 k ) ) )
3332, 12fvmptg 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
) ) )
3422, 30, 33syl2anc 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 ) )
3736biimpri 133 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
3837adantl 277 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>=
`  0 ) )  ->  k  e.  NN0 )
3935, 38, 18syl2anc 411 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>=
`  0 ) )  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
4022, 15eleqtrdi 2282 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
4135, 38, 8syl2anc 411 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>=
`  0 ) )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
4239, 40, 41fsum3ser 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 ) )
4334, 42eqtrd 2222 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( F `  j
)  =  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  j ) )
4414, 21, 43eqfnfvd 5640 . 2  |-  ( A  e.  CC  ->  F  =  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) )
4517efcvg 11715 . 2  |-  ( A  e.  CC  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
4644, 45eqbrtrd 4043 1  |-  ( A  e.  CC  ->  F  ~~>  ( exp `  A ) )
Colors of variables: wff set class
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)
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