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Theorem efsep 12406
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 2296 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
3 efsep.3 . . . . . . . 8  |-  M  e. 
NN0
43nn0zi 10064 . . . . . . 7  |-  M  e.  ZZ
54a1i 10 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2297 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
7 eluznn0 10304 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
83, 7mpan 651 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  NN0 )
9 efsep.1 . . . . . . . . . 10  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
109eftval 12374 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
1110adantl 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
12 efsep.4 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
13 eftcl 12371 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1412, 13sylan 457 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  CC )
1511, 14eqeltrd 2370 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
168, 15sylan2 460 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
179eftlcvg 12402 . . . . . . 7  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )
1812, 3, 17sylancl 643 . . . . . 6  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
192, 5, 6, 16, 18isum1p 12316 . . . . 5  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( F `
 M )  + 
sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) ) )
209eftval 12374 . . . . . . 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 2300 . . . . . . . 8  |-  ( M  +  1 )  =  N
2423fveq2i 5544 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  N )
2524sumeq1i 12187 . . . . . 6  |-  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) ( F `  k
)  =  sum_ k  e.  ( ZZ>= `  N )
( F `  k
)
2621, 25oveq12i 5886 . . . . 5  |-  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) )  =  ( ( ( A ^ M
)  /  ( ! `
 M ) )  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) )
2719, 26syl6eq 2344 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( ( A ^ M )  /  ( ! `  M ) )  + 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
2827oveq2d 5890 . . 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 12371 . . . . 5  |-  ( ( A  e.  CC  /\  M  e.  NN0 )  -> 
( ( A ^ M )  /  ( ! `  M )
)  e.  CC )
3112, 3, 30sylancl 643 . . . 4  |-  ( ph  ->  ( ( A ^ M )  /  ( ! `  M )
)  e.  CC )
32 peano2nn0 10020 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
333, 32ax-mp 8 . . . . . 6  |-  ( M  +  1 )  e. 
NN0
3422, 33eqeltri 2366 . . . . 5  |-  N  e. 
NN0
359eftlcl 12403 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k )  e.  CC )
3612, 34, 35sylancl 643 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k )  e.  CC )
3729, 31, 36addassd 8873 . . 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 2331 . 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 5889 . 2  |-  ( ph  ->  ( ( B  +  ( ( A ^ M )  /  ( ! `  M )
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) )
411, 38, 403eqtrd 2332 1  |-  ( ph  ->  ( exp `  A
)  =  ( D  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    / cdiv 9439   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062   ^cexp 11120   !cfa 11304    ~~> cli 11974   sum_csu 12174   expce 12359
This theorem is referenced by:  ef4p  12409  dveflem  19342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175
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