MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efsep Structured version   Unicode version

Theorem efsep 12703
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 2435 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
3 efsep.3 . . . . . . . 8  |-  M  e. 
NN0
43nn0zi 10298 . . . . . . 7  |-  M  e.  ZZ
54a1i 11 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2436 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
7 eluznn0 10538 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
83, 7mpan 652 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  NN0 )
9 efsep.1 . . . . . . . . . 10  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
109eftval 12671 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
1110adantl 453 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
12 efsep.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
13 eftcl 12668 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1412, 13sylan 458 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
1511, 14eqeltrd 2509 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
168, 15sylan2 461 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
179eftlcvg 12699 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
1812, 3, 17sylancl 644 . . . . . 6  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
192, 5, 6, 16, 18isum1p 12613 . . . . 5  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( F `
 M )  + 
sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) ) )
209eftval 12671 . . . . . . 7  |-  ( M  e.  NN0  ->  ( F `
 M )  =  ( ( A ^ M )  /  ( ! `  M )
) )
213, 20ax-mp 8 . . . . . 6  |-  ( F `
 M )  =  ( ( A ^ M )  /  ( ! `  M )
)
22 efsep.2 . . . . . . . . 9  |-  N  =  ( M  +  1 )
2322eqcomi 2439 . . . . . . . 8  |-  ( M  +  1 )  =  N
2423fveq2i 5723 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  N )
2524sumeq1i 12484 . . . . . 6  |-  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) ( F `  k
)  =  sum_ k  e.  ( ZZ>= `  N )
( F `  k
)
2621, 25oveq12i 6085 . . . . 5  |-  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) )  =  ( ( ( A ^ M
)  /  ( ! `
 M ) )  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) )
2719, 26syl6eq 2483 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( ( A ^ M )  /  ( ! `  M ) )  + 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
2827oveq2d 6089 . . 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 12668 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  /  ( ! `  M )
)  e.  CC )
3112, 3, 30sylancl 644 . . . 4  |-  ( ph  ->  ( ( A ^ M )  /  ( ! `  M )
)  e.  CC )
32 peano2nn0 10252 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
333, 32ax-mp 8 . . . . . 6  |-  ( M  +  1 )  e. 
NN0
3422, 33eqeltri 2505 . . . . 5  |-  N  e. 
NN0
359eftlcl 12700 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k )  e.  CC )
3612, 34, 35sylancl 644 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k )  e.  CC )
3729, 31, 36addassd 9102 . . 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 2470 . 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 6088 . 2  |-  ( ph  ->  ( ( B  +  ( ( A ^ M )  /  ( ! `  M )
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) )
411, 38, 403eqtrd 2471 1  |-  ( ph  ->  ( exp `  A
)  =  ( D  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   dom cdm 4870   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    + caddc 8985    / cdiv 9669   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480    seq cseq 11315   ^cexp 11374   !cfa 11558    ~~> cli 12270   sum_csu 12471   expce 12656
This theorem is referenced by:  ef4p  12706  dveflem  19855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-fac 11559  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472
  Copyright terms: Public domain W3C validator