Theorem fsumm1 12208

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 9947	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
4		fzsn 10489	. . . . . 6 |- ( N e. ZZ -> ( N ... N ) = { N } )
5	3, 4	syl 14	. . . . 5 |- ( ph -> ( N ... N ) = { N } )
6	5	ineq2d 3432	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = ( ( M ... ( N - 1 ) ) i^i { N } ) )
7	3	zred 9779	. . . . . 6 |- ( ph -> N e. RR )
8	7	ltm1d 9270	. . . . 5 |- ( ph -> ( N - 1 ) < N )
9		fzdisj 10474	. . . . 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 2273	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9942	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	1, 12	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
14		peano2zm 9693	. . . . . . . 8 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . . 7 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9780	. . . . . . . . . 10 |- ( ph -> M e. CC )
17		ax-1cn 8278	. . . . . . . . . 10 |- 1 e. CC
18		npcan 8542	. . . . . . . . . 10 |- ( ( M e. CC /\ 1 e. CC ) -> ( ( M - 1 ) + 1 ) = M )
19	16, 17, 18	sylancl 417	. . . . . . . . 9 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
20	19	fveq2d 5702	. . . . . . . 8 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
21	1, 20	eleqtrrd 2318	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
22		eluzp1m1 9962	. . . . . . 7 |- ( ( ( M - 1 ) e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
23	15, 21, 22	syl2anc 415	. . . . . 6 |- ( ph -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
24		fzsuc2 10503	. . . . . 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 415	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
26	3	zcnd 9780	. . . . . . 7 |- ( ph -> N e. CC )
27		npcan 8542	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
28	26, 17, 27	sylancl 417	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
29	28	oveq2d 6104	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	25, 29	eqtr3d 2273	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( M ... N ) )
31	28	sneqd 3722	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
32	31	uneq2d 3383	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
33	30, 32	eqtr3d 2273	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
34	13, 3	fzfigd 10889	. . 3 |- ( ph -> ( M ... N ) e. Fin )
35		fsumm1.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
36	11, 33, 34, 35	fsumsplit 12199	. 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 2307	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
39	35	ralrimiva 2623	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
40		eluzfz2 10453	. . . . . 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 2934	. . . 4 |- ( ph -> B e. CC )
43	37	sumsn 12203	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ B e. CC ) -> sum_ k e. { N } A = B )
44	1, 42, 43	syl2anc 415	. . 3 |- ( ph -> sum_ k e. { N } A = B )
45	44	oveq2d 6104	. 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 2271	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3709   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   1c1 8186   + caddc 8188   < clt 8366   - cmin 8504   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428   sum_csu 12144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145

This theorem is used by:  fzosump1  12209  fsump1  12212  telfsumo  12258  fsumparts  12262  binom1dif  12279  log2ublem2  16222  1sgmprm  16288  chtublem  16295