Theorem fsumm1 12057

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 9826	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
4		fzsn 10363	. . . . . 6 |- ( N e. ZZ -> ( N ... N ) = { N } )
5	3, 4	syl 14	. . . . 5 |- ( ph -> ( N ... N ) = { N } )
6	5	ineq2d 3410	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = ( ( M ... ( N - 1 ) ) i^i { N } ) )
7	3	zred 9663	. . . . . 6 |- ( ph -> N e. RR )
8	7	ltm1d 9171	. . . . 5 |- ( ph -> ( N - 1 ) < N )
9		fzdisj 10349	. . . . 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 2266	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9821	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	1, 12	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
14		peano2zm 9578	. . . . . . . 8 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . . 7 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9664	. . . . . . . . . 10 |- ( ph -> M e. CC )
17		ax-1cn 8185	. . . . . . . . . 10 |- 1 e. CC
18		npcan 8447	. . . . . . . . . 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 5652	. . . . . . . 8 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
21	1, 20	eleqtrrd 2311	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
22		eluzp1m1 9841	. . . . . . 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 10376	. . . . . 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 9664	. . . . . . 7 |- ( ph -> N e. CC )
27		npcan 8447	. . . . . . 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 6044	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	25, 29	eqtr3d 2266	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( M ... N ) )
31	28	sneqd 3686	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
32	31	uneq2d 3363	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
33	30, 32	eqtr3d 2266	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
34	13, 3	fzfigd 10756	. . 3 |- ( ph -> ( M ... N ) e. Fin )
35		fsumm1.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
36	11, 33, 34, 35	fsumsplit 12048	. 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 2300	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
39	35	ralrimiva 2606	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
40		eluzfz2 10329	. . . . . 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 2916	. . . 4 |- ( ph -> B e. CC )
43	37	sumsn 12052	. . . 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 6044	. 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 2264	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   u. cun 3199   i^i cin 3200   (/)c0 3496   {csn 3673   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   < clt 8273   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305   sum_csu 11993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994

This theorem is referenced by:  fzosump1  12058  fsump1  12061  telfsumo  12107  fsumparts  12111  binom1dif  12128  1sgmprm  15808