Theorem efsep 11701

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

Step	Hyp	Ref	Expression
1		efsep.6	. 2 |- ( ph -> ( exp ` A ) = ( B + sum_ k e. ( ZZ>= ` M ) ( F ` k ) ) )
2		eqid 2177	. . . . . 6 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
3		efsep.3	. . . . . . . 8 |- M e. NN0
4	3	nn0zi 9277	. . . . . . 7 |- M e. ZZ
5	4	a1i 9	. . . . . 6 |- ( ph -> M e. ZZ )
6		eqidd 2178	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( F ` k ) )
7		eluznn0 9601	. . . . . . . 8 |- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
8	3, 7	mpan 424	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. NN0 )
9		efsep.4	. . . . . . . . 9 |- ( ph -> A e. CC )
10		efsep.1	. . . . . . . . . 10 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
11	10	eftvalcn 11667	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
12	9, 11	sylan 283	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
13		eftcl 11664	. . . . . . . . 9 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
14	9, 13	sylan 283	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
15	12, 14	eqeltrd 2254	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
16	8, 15	sylan2 286	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
17	10	eftlcvg 11697	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> seq M ( + , F ) e. dom ~~> )
18	9, 3, 17	sylancl 413	. . . . . 6 |- ( ph -> seq M ( + , F ) e. dom ~~> )
19	2, 5, 6, 16, 18	isum1p 11502	. . . . 5 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( F ` k ) = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) ( F ` k ) ) )
20	10	eftvalcn 11667	. . . . . . 7 |- ( ( A e. CC /\ M e. NN0 ) -> ( F ` M ) = ( ( A ^ M ) / ( ! ` M ) ) )
21	9, 3, 20	sylancl 413	. . . . . 6 |- ( ph -> ( F ` M ) = ( ( A ^ M ) / ( ! ` M ) ) )
22		efsep.2	. . . . . . . . . 10 |- N = ( M + 1 )
23	22	eqcomi 2181	. . . . . . . . 9 |- ( M + 1 ) = N
24	23	fveq2i 5520	. . . . . . . 8 |- ( ZZ>= ` ( M + 1 ) ) = ( ZZ>= ` N )
25	24	sumeq1i 11373	. . . . . . 7 |- sum_ k e. ( ZZ>= ` ( M + 1 ) ) ( F ` k ) = sum_ k e. ( ZZ>= ` N ) ( F ` k )
26	25	a1i 9	. . . . . 6 |- ( ph -> sum_ k e. ( ZZ>= ` ( M + 1 ) ) ( F ` k ) = sum_ k e. ( ZZ>= ` N ) ( F ` k ) )
27	21, 26	oveq12d 5895	. . . . 5 |- ( ph -> ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) ( F ` k ) ) = ( ( ( A ^ M ) / ( ! ` M ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) )
28	19, 27	eqtrd 2210	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` M ) ( F ` k ) = ( ( ( A ^ M ) / ( ! ` M ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) )
29	28	oveq2d 5893	. . 3 |- ( ph -> ( B + sum_ k e. ( ZZ>= ` M ) ( F ` k ) ) = ( B + ( ( ( A ^ M ) / ( ! ` M ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) ) )
30		efsep.5	. . . 4 |- ( ph -> B e. CC )
31		eftcl 11664	. . . . 5 |- ( ( A e. CC /\ M e. NN0 ) -> ( ( A ^ M ) / ( ! ` M ) ) e. CC )
32	9, 3, 31	sylancl 413	. . . 4 |- ( ph -> ( ( A ^ M ) / ( ! ` M ) ) e. CC )
33		peano2nn0 9218	. . . . . . 7 |- ( M e. NN0 -> ( M + 1 ) e. NN0 )
34	3, 33	ax-mp 5	. . . . . 6 |- ( M + 1 ) e. NN0
35	22, 34	eqeltri 2250	. . . . 5 |- N e. NN0
36	10	eftlcl 11698	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( ZZ>= ` N ) ( F ` k ) e. CC )
37	9, 35, 36	sylancl 413	. . . 4 |- ( ph -> sum_ k e. ( ZZ>= ` N ) ( F ` k ) e. CC )
38	30, 32, 37	addassd 7982	. . 3 |- ( ph -> ( ( B + ( ( A ^ M ) / ( ! ` M ) ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) = ( B + ( ( ( A ^ M ) / ( ! ` M ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) ) )
39	29, 38	eqtr4d 2213	. 2 |- ( ph -> ( B + sum_ k e. ( ZZ>= ` M ) ( F ` k ) ) = ( ( B + ( ( A ^ M ) / ( ! ` M ) ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) )
40		efsep.7	. . 3 |- ( ph -> ( B + ( ( A ^ M ) / ( ! ` M ) ) ) = D )
41	40	oveq1d 5892	. 2 |- ( ph -> ( ( B + ( ( A ^ M ) / ( ! ` M ) ) ) + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) = ( D + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) )
42	1, 39, 41	3eqtrd 2214	1 |- ( ph -> ( exp ` A ) = ( D + sum_ k e. ( ZZ>= ` N ) ( F ` k ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   |-> cmpt 4066   dom cdm 4628   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   + caddc 7816   / cdiv 8631   NN0cn0 9178   ZZcz 9255   ZZ>=cuz 9530   seqcseq 10447   ^cexp 10521   !cfa 10707   ~~> cli 11288   sum_csu 11363   expce 11652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-fac 10708  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364

This theorem is referenced by:  ef4p  11704  dveflem  14226