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

Theorem efsep 12638
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 2387 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
3 efsep.3 . . . . . . . 8  |-  M  e. 
NN0
43nn0zi 10238 . . . . . . 7  |-  M  e.  ZZ
54a1i 11 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2388 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
7 eluznn0 10478 . . . . . . . 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 12606 . . . . . . . . 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 12603 . . . . . . . . 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 2461 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
168, 15sylan2 461 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
179eftlcvg 12634 . . . . . . 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 12548 . . . . 5  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( F `
 M )  + 
sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) ) )
209eftval 12606 . . . . . . 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 2391 . . . . . . . 8  |-  ( M  +  1 )  =  N
2423fveq2i 5671 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  N )
2524sumeq1i 12419 . . . . . 6  |-  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) ( F `  k
)  =  sum_ k  e.  ( ZZ>= `  N )
( F `  k
)
2621, 25oveq12i 6032 . . . . 5  |-  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) )  =  ( ( ( A ^ M
)  /  ( ! `
 M ) )  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) )
2719, 26syl6eq 2435 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( ( A ^ M )  /  ( ! `  M ) )  + 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
2827oveq2d 6036 . . 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 12603 . . . . 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 10192 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
333, 32ax-mp 8 . . . . . 6  |-  ( M  +  1 )  e. 
NN0
3422, 33eqeltri 2457 . . . . 5  |-  N  e. 
NN0
359eftlcl 12635 . . . . 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 9043 . . 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 2422 . 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 6035 . 2  |-  ( ph  ->  ( ( B  +  ( ( A ^ M )  /  ( ! `  M )
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) )
411, 38, 403eqtrd 2423 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 1649    e. wcel 1717    e. cmpt 4207   dom cdm 4818   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    / cdiv 9609   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420    seq cseq 11250   ^cexp 11309   !cfa 11493    ~~> cli 12205   sum_csu 12406   expce 12591
This theorem is referenced by:  ef4p  12641  dveflem  19730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-fac 11494  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407
  Copyright terms: Public domain W3C validator