Theorem fsumm1 11586

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 9615	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
4		fzsn 10146	. . . . . 6 |- ( N e. ZZ -> ( N ... N ) = { N } )
5	3, 4	syl 14	. . . . 5 |- ( ph -> ( N ... N ) = { N } )
6	5	ineq2d 3365	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = ( ( M ... ( N - 1 ) ) i^i { N } ) )
7	3	zred 9453	. . . . . 6 |- ( ph -> N e. RR )
8	7	ltm1d 8964	. . . . 5 |- ( ph -> ( N - 1 ) < N )
9		fzdisj 10132	. . . . 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 2231	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9611	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	1, 12	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
14		peano2zm 9369	. . . . . . . 8 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . . 7 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9454	. . . . . . . . . 10 |- ( ph -> M e. CC )
17		ax-1cn 7977	. . . . . . . . . 10 |- 1 e. CC
18		npcan 8240	. . . . . . . . . 10 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M )
19	16, 17, 18	sylancl 413	. . . . . . . . 9 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
20	19	fveq2d 5565	. . . . . . . 8 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
21	1, 20	eleqtrrd 2276	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
22		eluzp1m1 9630	. . . . . . 7 |- ( ( ( M - 1 ) e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
23	15, 21, 22	syl2anc 411	. . . . . 6 |- ( ph -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
24		fzsuc2 10159	. . . . . 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 411	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
26	3	zcnd 9454	. . . . . . 7 |- ( ph -> N e. CC )
27		npcan 8240	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
28	26, 17, 27	sylancl 413	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
29	28	oveq2d 5941	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	25, 29	eqtr3d 2231	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( M ... N ) )
31	28	sneqd 3636	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
32	31	uneq2d 3318	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
33	30, 32	eqtr3d 2231	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
34	13, 3	fzfigd 10528	. . 3 |- ( ph -> ( M ... N ) e. Fin )
35		fsumm1.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
36	11, 33, 34, 35	fsumsplit 11577	. 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 2265	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
39	35	ralrimiva 2570	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
40		eluzfz2 10112	. . . . . 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 2873	. . . 4 |- ( ph -> B e. CC )
43	37	sumsn 11581	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ B e. CC ) -> sum_ k e. { N } A = B )
44	1, 42, 43	syl2anc 411	. . 3 |- ( ph -> sum_ k e. { N } A = B )
45	44	oveq2d 5941	. 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 2229	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   u. cun 3155   i^i cin 3156   (/)c0 3451   {csn 3623   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7882   1c1 7885   + caddc 7887   < clt 8066   - cmin 8202   ZZcz 9331   ZZ>=cuz 9606   ...cfz 10088   sum_csu 11523

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7975  ax-resscn 7976  ax-1cn 7977  ax-1re 7978  ax-icn 7979  ax-addcl 7980  ax-addrcl 7981  ax-mulcl 7982  ax-mulrcl 7983  ax-addcom 7984  ax-mulcom 7985  ax-addass 7986  ax-mulass 7987  ax-distr 7988  ax-i2m1 7989  ax-0lt1 7990  ax-1rid 7991  ax-0id 7992  ax-rnegex 7993  ax-precex 7994  ax-cnre 7995  ax-pre-ltirr 7996  ax-pre-ltwlin 7997  ax-pre-lttrn 7998  ax-pre-apti 7999  ax-pre-ltadd 8000  ax-pre-mulgt0 8001  ax-pre-mulext 8002  ax-arch 8003  ax-caucvg 8004

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6202  df-2nd 6203  df-recs 6367  df-irdg 6432  df-frec 6453  df-1o 6478  df-oadd 6482  df-er 6596  df-en 6804  df-dom 6805  df-fin 6806  df-pnf 8068  df-mnf 8069  df-xr 8070  df-ltxr 8071  df-le 8072  df-sub 8204  df-neg 8205  df-reap 8607  df-ap 8614  df-div 8705  df-inn 8996  df-2 9054  df-3 9055  df-4 9056  df-n0 9255  df-z 9332  df-uz 9607  df-q 9699  df-rp 9734  df-fz 10089  df-fzo 10223  df-seqfrec 10545  df-exp 10636  df-ihash 10873  df-cj 11012  df-re 11013  df-im 11014  df-rsqrt 11168  df-abs 11169  df-clim 11449  df-sumdc 11524

This theorem is referenced by:  fzosump1  11587  fsump1  11590  telfsumo  11636  fsumparts  11640  binom1dif  11657  1sgmprm  15277