Theorem fsumm1 10810

Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.)

Hypotheses

Ref	Expression
fsumm1.1	|- ( ph -> N e. ( ZZ>= ` M ) )
fsumm1.2	|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
fsumm1.3	|- ( k = N -> A = B )

Assertion

Ref	Expression
fsumm1	|- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )

Distinct variable groups:   B, k   k, M   k, N   ph, k

Allowed substitution hint:   A( k)

Proof of Theorem fsumm1

Step	Hyp	Ref	Expression
1		fsumm1.1	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzelz 9028	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
4		fzsn 9480	. . . . . 6 |- ( N e. ZZ -> ( N ... N ) = { N } )
5	3, 4	syl 14	. . . . 5 |- ( ph -> ( N ... N ) = { N } )
6	5	ineq2d 3201	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = ( ( M ... ( N - 1 ) ) i^i { N } ) )
7	3	zred 8868	. . . . . 6 |- ( ph -> N e. RR )
8	7	ltm1d 8393	. . . . 5 |- ( ph -> ( N - 1 ) < N )
9		fzdisj 9466	. . . . 5 |- ( ( N - 1 ) < N -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = (/) )
10	8, 9	syl 14	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = (/) )
11	6, 10	eqtr3d 2122	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9024	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	1, 12	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
14		peano2zm 8788	. . . . . . . 8 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . . 7 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 8869	. . . . . . . . . 10 |- ( ph -> M e. CC )
17		ax-1cn 7438	. . . . . . . . . 10 |- 1 e. CC
18		npcan 7691	. . . . . . . . . 10 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M )
19	16, 17, 18	sylancl 404	. . . . . . . . 9 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
20	19	fveq2d 5309	. . . . . . . 8 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
21	1, 20	eleqtrrd 2167	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
22		eluzp1m1 9042	. . . . . . 7 |- ( ( ( M - 1 ) e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
23	15, 21, 22	syl2anc 403	. . . . . 6 |- ( ph -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
24		fzsuc2 9493	. . . . . 6 |- ( ( M e. ZZ /\ ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
25	13, 23, 24	syl2anc 403	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
26	3	zcnd 8869	. . . . . . 7 |- ( ph -> N e. CC )
27		npcan 7691	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
28	26, 17, 27	sylancl 404	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
29	28	oveq2d 5668	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	25, 29	eqtr3d 2122	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( M ... N ) )
31	28	sneqd 3459	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
32	31	uneq2d 3154	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
33	30, 32	eqtr3d 2122	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
34	13, 3	fzfigd 9838	. . 3 |- ( ph -> ( M ... N ) e. Fin )
35		fsumm1.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
36	11, 33, 34, 35	fsumsplit 10801	. 2 |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. { N } A ) )
37		fsumm1.3	. . . . . 6 |- ( k = N -> A = B )
38	37	eleq1d 2156	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
39	35	ralrimiva 2446	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
40		eluzfz2 9446	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
41	1, 40	syl 14	. . . . 5 |- ( ph -> N e. ( M ... N ) )
42	38, 39, 41	rspcdva 2727	. . . 4 |- ( ph -> B e. CC )
43	37	sumsn 10805	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ B e. CC ) -> sum_ k e. { N } A = B )
44	1, 42, 43	syl2anc 403	. . 3 |- ( ph -> sum_ k e. { N } A = B )
45	44	oveq2d 5668	. 2 |- ( ph -> ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. { N } A ) = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )
46	36, 45	eqtrd 2120	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   u. cun 2997   i^i cin 2998   (/)c0 3286   {csn 3446   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7348   1c1 7351   + caddc 7353   < clt 7522   - cmin 7653   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424   sum_csu 10742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743

This theorem is referenced by:  fzosump1  10811  fsump1  10814  telfsumo  10860  fsumparts  10864  binom1dif  10881