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Theorem efcvgfsum 14118
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 oveq2 6305 . . . . . . . 8  |-  ( n  =  j  ->  (
0 ... n )  =  ( 0 ... j
) )
21sumeq1d 13745 . . . . . . 7  |-  ( n  =  j  ->  sum_ k  e.  ( 0 ... n
) ( ( A ^ k )  / 
( ! `  k
) )  =  sum_ k  e.  ( 0 ... j ) ( ( A ^ k
)  /  ( ! `
 k ) ) )
3 efcvgfsum.1 . . . . . . 7  |-  F  =  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A ^ k
)  /  ( ! `
 k ) ) )
4 sumex 13732 . . . . . . 7  |-  sum_ k  e.  ( 0 ... j
) ( ( A ^ k )  / 
( ! `  k
) )  e.  _V
52, 3, 4fvmpt 5956 . . . . . 6  |-  ( j  e.  NN0  ->  ( F `
 j )  = 
sum_ k  e.  ( 0 ... j ) ( ( A ^
k )  /  ( ! `  k )
) )
65adantl 467 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( F `  j
)  =  sum_ k  e.  ( 0 ... j
) ( ( A ^ k )  / 
( ! `  k
) ) )
7 elfznn0 11881 . . . . . . . 8  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
87adantl 467 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  k  e.  NN0 )
9 eqid 2420 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
109eftval 14109 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
118, 10syl 17 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
12 simpr 462 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  NN0 )
13 nn0uz 11189 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1412, 13syl6eleq 2518 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
15 simpll 758 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  A  e.  CC )
16 eftcl 14106 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1715, 8, 16syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
1811, 14, 17fsumser 13774 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j ) ( ( A ^ k
)  /  ( ! `
 k ) )  =  (  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  j ) )
196, 18eqtrd 2461 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( F `  j
)  =  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  j ) )
2019ralrimiva 2837 . . 3  |-  ( A  e.  CC  ->  A. j  e.  NN0  ( F `  j )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 j ) )
21 sumex 13732 . . . . 5  |-  sum_ k  e.  ( 0 ... n
) ( ( A ^ k )  / 
( ! `  k
) )  e.  _V
2221, 3fnmpti 5716 . . . 4  |-  F  Fn  NN0
23 0z 10944 . . . . . 6  |-  0  e.  ZZ
24 seqfn 12218 . . . . . 6  |-  ( 0  e.  ZZ  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  Fn  ( ZZ>= `  0 )
)
2523, 24ax-mp 5 . . . . 5  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) )  Fn  ( ZZ>= ` 
0 )
2613fneq2i 5681 . . . . 5  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  Fn 
NN0 
<->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) )  Fn  ( ZZ>= `  0
) )
2725, 26mpbir 212 . . . 4  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) )  Fn  NN0
28 eqfnfv 5983 . . . 4  |-  ( ( F  Fn  NN0  /\  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  Fn 
NN0 )  ->  ( F  =  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) )  <->  A. j  e.  NN0  ( F `  j )  =  (  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  j ) ) )
2922, 27, 28mp2an 676 . . 3  |-  ( F  =  seq 0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) )  <->  A. j  e.  NN0  ( F `  j )  =  (  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  j ) )
3020, 29sylibr 215 . 2  |-  ( A  e.  CC  ->  F  =  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) )
319efcvg 14117 . 2  |-  ( A  e.  CC  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
3230, 31eqbrtrd 4438 1  |-  ( A  e.  CC  ->  F  ~~>  ( exp `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   class class class wbr 4417    |-> cmpt 4476    Fn wfn 5588   ` cfv 5593  (class class class)co 6297   CCcc 9533   0cc0 9535    + caddc 9538    / cdiv 10265   NN0cn0 10865   ZZcz 10933   ZZ>=cuz 11155   ...cfz 11778    seqcseq 12206   ^cexp 12265   !cfa 12452    ~~> cli 13526   sum_csu 13730   expce 14092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-inf2 8144  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612  ax-pre-sup 9613  ax-addf 9614  ax-mulf 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-se 4806  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7954  df-inf 7955  df-oi 8023  df-card 8370  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-div 10266  df-nn 10606  df-2 10664  df-3 10665  df-n0 10866  df-z 10934  df-uz 11156  df-rp 11299  df-ico 11637  df-fz 11779  df-fzo 11910  df-fl 12021  df-seq 12207  df-exp 12266  df-fac 12453  df-hash 12509  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099
This theorem is referenced by: (None)
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