Theorem fsum3ser 11741

Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 11756 and fsump1 11764, which should make our notation clear and from which, along with closure fsumcl 11744, 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 2205	. . . . 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 2266	. . . . . 6 |- ( m = k -> ( m e. ( M ... N ) <-> k e. ( M ... N ) ) )
3		fveq2 5578	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
4	2, 3	ifbieq1d 3593	. . . . 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 2282	. . . . . . 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 8067	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. ( M ... N ) ) -> 0 e. CC )
11		eluzelz 9659	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
12		eluzel2 9655	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> M e. ZZ )
13		fsum3ser.2	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
14		eluzelz 9659	. . . . . . . . 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 10164	. . . . . . 7 |- ( ( k e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID k e. ( M ... N ) )
18	11, 12, 16, 17	syl2an23an 1312	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. ( M ... N ) )
19	9, 10, 18	ifcldadc 3600	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) e. CC )
20	1, 4, 5, 19	fvmptd3 5675	. . . 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 3588	. . . 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 2238	. . 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 10145	. . . 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 3214	. . 3 |- ( ph -> ( M ... N ) C_ ( M ... N ) )
26	22, 13, 24, 18, 25	fsumsersdc 11739	. 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 3576	. . . . 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 2238	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
31		eleq1w 2266	. . . . . 6 |- ( m = x -> ( m e. ( M ... N ) <-> x e. ( M ... N ) ) )
32		fveq2 5578	. . . . . 6 |- ( m = x -> ( F ` m ) = ( F ` x ) )
33	31, 32	ifbieq1d 3593	. . . . 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 5578	. . . . . . . 8 |- ( k = x -> ( F ` k ) = ( F ` x ) )
36	35	eleq1d 2274	. . . . . . 7 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
37	8	ralrimiva 2579	. . . . . . . 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 2882	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
40		0cnd 8067	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 e. CC )
41		eluzelz 9659	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
42		eluzel2 9655	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> M e. ZZ )
43	15	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> N e. ZZ )
44		fzdcel 10164	. . . . . . 7 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID x e. ( M ... N ) )
45	41, 42, 43, 44	syl2an23an 1312	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> DECID x e. ( M ... N ) )
46	39, 40, 45	ifcldcd 3608	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> if ( x e. ( M ... N ) , ( F ` x ) , 0 ) e. CC )
47	1, 33, 34, 46	fvmptd3 5675	. . . 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 2282	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` x ) e. CC )
49	36	cbvralv 2738	. . . . 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 2594	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
52		addcl 8052	. . . 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 10626	. 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 2238	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 836   = wceq 1373   e. wcel 2176   A.wral 2484   ifcif 3571   |-> cmpt 4106   ` cfv 5272  (class class class)co 5946   CCcc 7925   0cc0 7927   + caddc 7930   ZZcz 9374   ZZ>=cuz 9650   ...cfz 10132   seqcseq 10594   sum_csu 11697

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-ihash 10923  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698

This theorem is referenced by:  isumclim3  11767  iserabs  11819  isumsplit  11835  trireciplem  11844  geolim  11855  geo2lim  11860  cvgratnnlemseq  11870  mertenslem2  11880  mertensabs  11881  efcvgfsum  12011  effsumlt  12036  cvgcmp2nlemabs  16008