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Theorem efcvgfsum 12690
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 6091 . . . . . . . 8  |-  ( n  =  j  ->  (
0 ... n )  =  ( 0 ... j
) )
21sumeq1d 12497 . . . . . . 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 12483 . . . . . . 7  |-  sum_ k  e.  ( 0 ... j
) ( ( A ^ k )  / 
( ! `  k
) )  e.  _V
52, 3, 4fvmpt 5808 . . . . . 6  |-  ( j  e.  NN0  ->  ( F `
 j )  = 
sum_ k  e.  ( 0 ... j ) ( ( A ^
k )  /  ( ! `  k )
) )
65adantl 454 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( F `  j
)  =  sum_ k  e.  ( 0 ... j
) ( ( A ^ k )  / 
( ! `  k
) ) )
7 elfznn0 11085 . . . . . . . 8  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
87adantl 454 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  k  e.  NN0 )
9 eqid 2438 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
109eftval 12681 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
118, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
12 simpr 449 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  NN0 )
13 nn0uz 10522 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
1412, 13syl6eleq 2528 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
15 simpll 732 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  A  e.  CC )
16 eftcl 12678 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1715, 8, 16syl2anc 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
1811, 14, 17fsumser 12526 . . . . 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 2470 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( F `  j
)  =  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  j ) )
2019ralrimiva 2791 . . 3  |-  ( A  e.  CC  ->  A. j  e.  NN0  ( F `  j )  =  (  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 j ) )
21 sumex 12483 . . . . 5  |-  sum_ k  e.  ( 0 ... n
) ( ( A ^ k )  / 
( ! `  k
) )  e.  _V
2221, 3fnmpti 5575 . . . 4  |-  F  Fn  NN0
23 0z 10295 . . . . . 6  |-  0  e.  ZZ
24 seqfn 11337 . . . . . 6  |-  ( 0  e.  ZZ  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  Fn  ( ZZ>= `  0 )
)
2523, 24ax-mp 8 . . . . 5  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) )  Fn  ( ZZ>= ` 
0 )
2613fneq2i 5542 . . . . 5  |-  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  Fn 
NN0 
<->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) )  Fn  ( ZZ>= `  0
) )
2725, 26mpbir 202 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) )  Fn  NN0
28 eqfnfv 5829 . . . 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 655 . . 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 205 . 2  |-  ( A  e.  CC  ->  F  =  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) )
319efcvg 12689 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
3230, 31eqbrtrd 4234 1  |-  ( A  e.  CC  ->  F  ~~>  ( exp `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4214    e. cmpt 4268    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    + caddc 8995    / cdiv 9679   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045    seq cseq 11325   ^cexp 11384   !cfa 11568    ~~> cli 12280   sum_csu 12481   expce 12666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-fac 11569  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672
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