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Theorem efsep 13695
Description: Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
Hypotheses
Ref Expression
efsep.1  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
efsep.2  |-  N  =  ( M  +  1 )
efsep.3  |-  M  e. 
NN0
efsep.4  |-  ( ph  ->  A  e.  CC )
efsep.5  |-  ( ph  ->  B  e.  CC )
efsep.6  |-  ( ph  ->  ( exp `  A
)  =  ( B  +  sum_ k  e.  (
ZZ>= `  M ) ( F `  k ) ) )
efsep.7  |-  ( ph  ->  ( B  +  ( ( A ^ M
)  /  ( ! `
 M ) ) )  =  D )
Assertion
Ref Expression
efsep  |-  ( ph  ->  ( exp `  A
)  =  ( D  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
Distinct variable groups:    k, n, A    k, F    k, M, n    k, N, n    ph, k
Allowed substitution hints:    ph( n)    B( k, n)    D( k, n)    F( n)

Proof of Theorem efsep
StepHypRef Expression
1 efsep.6 . 2  |-  ( ph  ->  ( exp `  A
)  =  ( B  +  sum_ k  e.  (
ZZ>= `  M ) ( F `  k ) ) )
2 eqid 2460 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
3 efsep.3 . . . . . . . 8  |-  M  e. 
NN0
43nn0zi 10878 . . . . . . 7  |-  M  e.  ZZ
54a1i 11 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2461 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
7 eluznn0 11140 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
83, 7mpan 670 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  NN0 )
9 efsep.1 . . . . . . . . . 10  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
109eftval 13663 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
1110adantl 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
12 efsep.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
13 eftcl 13660 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1412, 13sylan 471 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
1511, 14eqeltrd 2548 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
168, 15sylan2 474 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
179eftlcvg 13691 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  seq M (  +  ,  F )  e.  dom  ~~>  )
1812, 3, 17sylancl 662 . . . . . 6  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
192, 5, 6, 16, 18isum1p 13605 . . . . 5  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( F `
 M )  + 
sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) ) )
209eftval 13663 . . . . . . 7  |-  ( M  e.  NN0  ->  ( F `
 M )  =  ( ( A ^ M )  /  ( ! `  M )
) )
213, 20ax-mp 5 . . . . . 6  |-  ( F `
 M )  =  ( ( A ^ M )  /  ( ! `  M )
)
22 efsep.2 . . . . . . . . 9  |-  N  =  ( M  +  1 )
2322eqcomi 2473 . . . . . . . 8  |-  ( M  +  1 )  =  N
2423fveq2i 5860 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  N )
2524sumeq1i 13469 . . . . . 6  |-  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) ( F `  k
)  =  sum_ k  e.  ( ZZ>= `  N )
( F `  k
)
2621, 25oveq12i 6287 . . . . 5  |-  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) )  =  ( ( ( A ^ M
)  /  ( ! `
 M ) )  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) )
2719, 26syl6eq 2517 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( ( A ^ M )  /  ( ! `  M ) )  + 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
2827oveq2d 6291 . . 3  |-  ( ph  ->  ( B  +  sum_ k  e.  ( ZZ>= `  M ) ( F `
 k ) )  =  ( B  +  ( ( ( A ^ M )  / 
( ! `  M
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) ) )
29 efsep.5 . . . 4  |-  ( ph  ->  B  e.  CC )
30 eftcl 13660 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  /  ( ! `  M )
)  e.  CC )
3112, 3, 30sylancl 662 . . . 4  |-  ( ph  ->  ( ( A ^ M )  /  ( ! `  M )
)  e.  CC )
32 peano2nn0 10825 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
333, 32ax-mp 5 . . . . . 6  |-  ( M  +  1 )  e. 
NN0
3422, 33eqeltri 2544 . . . . 5  |-  N  e. 
NN0
359eftlcl 13692 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k )  e.  CC )
3612, 34, 35sylancl 662 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k )  e.  CC )
3729, 31, 36addassd 9607 . . 3  |-  ( ph  ->  ( ( B  +  ( ( A ^ M )  /  ( ! `  M )
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) )  =  ( B  +  ( ( ( A ^ M )  / 
( ! `  M
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) ) )
3828, 37eqtr4d 2504 . 2  |-  ( ph  ->  ( B  +  sum_ k  e.  ( ZZ>= `  M ) ( F `
 k ) )  =  ( ( B  +  ( ( A ^ M )  / 
( ! `  M
) ) )  + 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
39 efsep.7 . . 3  |-  ( ph  ->  ( B  +  ( ( A ^ M
)  /  ( ! `
 M ) ) )  =  D )
4039oveq1d 6290 . 2  |-  ( ph  ->  ( ( B  +  ( ( A ^ M )  /  ( ! `  M )
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) )
411, 38, 403eqtrd 2505 1  |-  ( ph  ->  ( exp `  A
)  =  ( D  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    |-> cmpt 4498   dom cdm 4992   ` cfv 5579  (class class class)co 6275   CCcc 9479   1c1 9482    + caddc 9484    / cdiv 10195   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071    seqcseq 12063   ^cexp 12122   !cfa 12308    ~~> cli 13256   sum_csu 13457   expce 13648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ico 11524  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-fac 12309  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458
This theorem is referenced by:  ef4p  13698  dveflem  22108
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