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Theorem efsep 12390
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 2283 . . . . . 6  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
3 efsep.3 . . . . . . . 8  |-  M  e. 
NN0
43nn0zi 10048 . . . . . . 7  |-  M  e.  ZZ
54a1i 10 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
6 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
7 eluznn0 10288 . . . . . . . 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 12358 . . . . . . . . 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 12355 . . . . . . . . 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 2357 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
168, 15sylan2 460 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
179eftlcvg 12386 . . . . . . 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 12300 . . . . 5  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( F `
 M )  + 
sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) ) )
209eftval 12358 . . . . . . 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 2287 . . . . . . . 8  |-  ( M  +  1 )  =  N
2423fveq2i 5528 . . . . . . 7  |-  ( ZZ>= `  ( M  +  1
) )  =  (
ZZ>= `  N )
2524sumeq1i 12171 . . . . . 6  |-  sum_ k  e.  ( ZZ>= `  ( M  +  1 ) ) ( F `  k
)  =  sum_ k  e.  ( ZZ>= `  N )
( F `  k
)
2621, 25oveq12i 5870 . . . . 5  |-  ( ( F `  M )  +  sum_ k  e.  (
ZZ>= `  ( M  + 
1 ) ) ( F `  k ) )  =  ( ( ( A ^ M
)  /  ( ! `
 M ) )  +  sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) )
2719, 26syl6eq 2331 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( F `  k )  =  ( ( ( A ^ M )  /  ( ! `  M ) )  + 
sum_ k  e.  (
ZZ>= `  N ) ( F `  k ) ) )
2827oveq2d 5874 . . 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 12355 . . . . 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 10004 . . . . . . 7  |-  ( M  e.  NN0  ->  ( M  +  1 )  e. 
NN0 )
333, 32ax-mp 8 . . . . . 6  |-  ( M  +  1 )  e. 
NN0
3422, 33eqeltri 2353 . . . . 5  |-  N  e. 
NN0
359eftlcl 12387 . . . . 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 8857 . . 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 2318 . 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 5873 . 2  |-  ( ph  ->  ( ( B  +  ( ( A ^ M )  /  ( ! `  M )
) )  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) )  =  ( D  +  sum_ k  e.  ( ZZ>= `  N ) ( F `
 k ) ) )
411, 38, 403eqtrd 2319 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 1623    e. wcel 1684    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    / cdiv 9423   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046   ^cexp 11104   !cfa 11288    ~~> cli 11958   sum_csu 12158   expce 12343
This theorem is referenced by:  ef4p  12393  dveflem  19326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159
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