Theorem fsum3ser 11423

Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11438 and fsump1 11446, which should make our notation clear and from which, along with closure fsumcl 11426, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)

Hypotheses

Ref	Expression
fsum3ser.1	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
fsum3ser.2	|- ( ph -> N e. ( ZZ>= ` M ) )
fsum3ser.3	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )

Assertion

Ref	Expression
fsum3ser	|- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

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

Allowed substitution hint:   A( k)

Proof of Theorem fsum3ser

Dummy variables m x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2189	. . . . 5 |- ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) = ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) )
2		eleq1w 2250	. . . . . 6 |- ( m = k -> ( m e. ( M ... N ) <-> k e. ( M ... N ) ) )
3		fveq2 5530	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
4	2, 3	ifbieq1d 3571	. . . . 5 |- ( m = k -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
5		simpr 110	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
6		fsum3ser.1	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
7		fsum3ser.3	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
8	6, 7	eqeltrd 2266	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
9	8	adantr 276	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
10		0cnd 7968	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. ( M ... N ) ) -> 0 e. CC )
11		eluzelz 9555	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
12		eluzel2 9551	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> M e. ZZ )
13		fsum3ser.2	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
14		eluzelz 9555	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
16	15	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> N e. ZZ )
17		fzdcel 10058	. . . . . . 7 |- ( ( k e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID k e. ( M ... N ) )
18	11, 12, 16, 17	syl2an23an 1310	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. ( M ... N ) )
19	9, 10, 18	ifcldadc 3578	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) e. CC )
20	1, 4, 5, 19	fvmptd3 5625	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
21	6	ifeq1d 3566	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = if ( k e. ( M ... N ) , A , 0 ) )
22	20, 21	eqtrd 2222	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , A , 0 ) )
23		elfzuz 10039	. . . 4 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
24	23, 7	sylan2 286	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
25		ssidd 3191	. . 3 |- ( ph -> ( M ... N ) C_ ( M ... N ) )
26	22, 13, 24, 18, 25	fsumsersdc 11421	. 2 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ) ` N ) )
27	23, 20	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
28		iftrue 3554	. . . . 5 |- ( k e. ( M ... N ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
29	28	adantl 277	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
30	27, 29	eqtrd 2222	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
31		eleq1w 2250	. . . . . 6 |- ( m = x -> ( m e. ( M ... N ) <-> x e. ( M ... N ) ) )
32		fveq2 5530	. . . . . 6 |- ( m = x -> ( F ` m ) = ( F ` x ) )
33	31, 32	ifbieq1d 3571	. . . . 5 |- ( m = x -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( x e. ( M ... N ) , ( F ` x ) , 0 ) )
34		simpr 110	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
35		fveq2 5530	. . . . . . . 8 |- ( k = x -> ( F ` k ) = ( F ` x ) )
36	35	eleq1d 2258	. . . . . . 7 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
37	8	ralrimiva 2563	. . . . . . . 8 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
38	37	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
39	36, 38, 34	rspcdva 2861	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
40		0cnd 7968	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 e. CC )
41		eluzelz 9555	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
42		eluzel2 9551	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> M e. ZZ )
43	15	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> N e. ZZ )
44		fzdcel 10058	. . . . . . 7 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID x e. ( M ... N ) )
45	41, 42, 43, 44	syl2an23an 1310	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> DECID x e. ( M ... N ) )
46	39, 40, 45	ifcldcd 3585	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> if ( x e. ( M ... N ) , ( F ` x ) , 0 ) e. CC )
47	1, 33, 34, 46	fvmptd3 5625	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` x ) = if ( x e. ( M ... N ) , ( F ` x ) , 0 ) )
48	47, 46	eqeltrd 2266	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` x ) e. CC )
49	36	cbvralv 2718	. . . . 5 |- ( A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC <-> A. x e. ( ZZ>= ` M ) ( F ` x ) e. CC )
50	37, 49	sylib 122	. . . 4 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. CC )
51	50	r19.21bi 2578	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
52		addcl 7954	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
53	52	adantl 277	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
54	13, 30, 48, 51, 53	seq3fveq 10489	. 2 |- ( ph -> ( seq M ( + , ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ) ` N ) = ( seq M ( + , F ) ` N ) )
55	26, 54	eqtrd 2222	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   e. wcel 2160   A.wral 2468   ifcif 3549   |-> cmpt 4079   ` cfv 5231  (class class class)co 5891   CCcc 7827   0cc0 7829   + caddc 7832   ZZcz 9271   ZZ>=cuz 9546   ...cfz 10026   seqcseq 10463   sum_csu 11379

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-fz 10027  df-fzo 10161  df-seqfrec 10464  df-exp 10538  df-ihash 10774  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-clim 11305  df-sumdc 11380

This theorem is referenced by:  isumclim3  11449  iserabs  11501  isumsplit  11517  trireciplem  11526  geolim  11537  geo2lim  11542  cvgratnnlemseq  11552  mertenslem2  11562  mertensabs  11563  efcvgfsum  11693  effsumlt  11718  cvgcmp2nlemabs  15165