Theorem fsumm1 11842

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 9692	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
4		fzsn 10223	. . . . . 6 |- ( N e. ZZ -> ( N ... N ) = { N } )
5	3, 4	syl 14	. . . . 5 |- ( ph -> ( N ... N ) = { N } )
6	5	ineq2d 3382	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i ( N ... N ) ) = ( ( M ... ( N - 1 ) ) i^i { N } ) )
7	3	zred 9530	. . . . . 6 |- ( ph -> N e. RR )
8	7	ltm1d 9040	. . . . 5 |- ( ph -> ( N - 1 ) < N )
9		fzdisj 10209	. . . . 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 2242	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9688	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	1, 12	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
14		peano2zm 9445	. . . . . . . 8 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . . 7 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9531	. . . . . . . . . 10 |- ( ph -> M e. CC )
17		ax-1cn 8053	. . . . . . . . . 10 |- 1 e. CC
18		npcan 8316	. . . . . . . . . 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 5603	. . . . . . . 8 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
21	1, 20	eleqtrrd 2287	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
22		eluzp1m1 9707	. . . . . . 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 10236	. . . . . 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 9531	. . . . . . 7 |- ( ph -> N e. CC )
27		npcan 8316	. . . . . . 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 5983	. . . . 5 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
30	25, 29	eqtr3d 2242	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( M ... N ) )
31	28	sneqd 3656	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
32	31	uneq2d 3335	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
33	30, 32	eqtr3d 2242	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
34	13, 3	fzfigd 10613	. . 3 |- ( ph -> ( M ... N ) e. Fin )
35		fsumm1.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
36	11, 33, 34, 35	fsumsplit 11833	. 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 2276	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
39	35	ralrimiva 2581	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
40		eluzfz2 10189	. . . . . 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 2889	. . . 4 |- ( ph -> B e. CC )
43	37	sumsn 11837	. . . 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 5983	. 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 2240	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... ( N - 1 ) ) A + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   u. cun 3172   i^i cin 3173   (/)c0 3468   {csn 3643   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   1c1 7961   + caddc 7963   < clt 8142   - cmin 8278   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   sum_csu 11779

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-ihash 10958  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-sumdc 11780

This theorem is referenced by:  fzosump1  11843  fsump1  11846  telfsumo  11892  fsumparts  11896  binom1dif  11913  1sgmprm  15581