Theorem fsum3cvg 12089

Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)

Hypotheses

Ref	Expression
isummo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
isummo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
isummo.dc	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
isumrb.3	|- ( ph -> N e. ( ZZ>= ` M ) )
fisumcvg.4	|- ( ph -> A C_ ( M ... N ) )

Assertion

Ref	Expression
fsum3cvg	|- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

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

Allowed substitution hint:   B( k)

Proof of Theorem fsum3cvg

Dummy variables n z m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		isumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9881	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. 2 |- ( ph -> N e. ZZ )
5		seqex 10835	. . 3 |- seq M ( + , F ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( + , F ) e. _V )
7		eqid 2234	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 9876	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 14	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 9881	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 277	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 3631	. . . . . . . . . . 11 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
13	12	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) = B )
14		isummo.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. CC )
15	13, 14	eqeltrd 2311	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
16	15	ex 115	. . . . . . . 8 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
17	16	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
18		iffalse 3634	. . . . . . . . 9 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
19		0cn 8282	. . . . . . . . 9 |- 0 e. CC
20	18, 19	eqeltrdi 2325	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
21	20	a1i 9	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( -. k e. A -> if ( k e. A , B , 0 ) e. CC ) )
22		isummo.dc	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
23		exmiddc 844	. . . . . . . 8 |- (DECID k e. A -> ( k e. A \/ -. k e. A ) )
24	22, 23	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A \/ -. k e. A ) )
25	17, 21, 24	mpjaod 726	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
26		isummo.1	. . . . . . 7 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
27	26	fvmpt2 5766	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 0 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
28	11, 25, 27	syl2anc 411	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
29	28, 25	eqeltrd 2311	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	7, 9, 29	serf 10869	. . 3 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> CC )
31	30, 2	ffvelcdmd 5818	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
32		addrid 8427	. . . . 5 |- ( m e. CC -> ( m + 0 ) = m )
33	32	adantl 277	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. CC ) -> ( m + 0 ) = m )
34	2	adantr 276	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
35		simpr 110	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> n e. ( ZZ>= ` N ) )
36	31	adantr 276	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` N ) e. CC )
37		elfzuz 10374	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
38		eluzelz 9881	. . . . . . . . 9 |- ( m e. ( ZZ>= ` ( N + 1 ) ) -> m e. ZZ )
39	38	adantl 277	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ZZ )
40		fisumcvg.4	. . . . . . . . . . . 12 |- ( ph -> A C_ ( M ... N ) )
41	40	sseld 3241	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
42		fznuz 10458	. . . . . . . . . . 11 |- ( m e. ( M ... N ) -> -. m e. ( ZZ>= ` ( N + 1 ) ) )
43	41, 42	syl6 33	. . . . . . . . . 10 |- ( ph -> ( m e. A -> -. m e. ( ZZ>= ` ( N + 1 ) ) ) )
44	43	con2d 629	. . . . . . . . 9 |- ( ph -> ( m e. ( ZZ>= ` ( N + 1 ) ) -> -. m e. A ) )
45	44	imp 124	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> -. m e. A )
46	39, 45	eldifd 3224	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
47		fveqeq2 5684	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 0 <-> ( F ` m ) = 0 ) )
48		eldifi 3345	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
49		eldifn 3346	. . . . . . . . . . . 12 |- ( k e. ( ZZ \ A ) -> -. k e. A )
50	49, 18	syl 14	. . . . . . . . . . 11 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
51	50, 19	eqeltrdi 2325	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
52	48, 51, 27	syl2anc 411	. . . . . . . . 9 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
53	52, 50	eqtrd 2267	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
54	47, 53	vtoclga 2883	. . . . . . 7 |- ( m e. ( ZZ \ A ) -> ( F ` m ) = 0 )
55	46, 54	syl 14	. . . . . 6 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` m ) = 0 )
56	37, 55	sylan2 286	. . . . 5 |- ( ( ph /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
57	56	adantlr 477	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
58		fveq2 5675	. . . . . 6 |- ( k = m -> ( F ` k ) = ( F ` m ) )
59	58	eleq1d 2303	. . . . 5 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
60	29	ralrimiva 2617	. . . . . 6 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
61	60	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
62		simpr 110	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> m e. ( ZZ>= ` M ) )
63	59, 61, 62	rspcdva 2928	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> ( F ` m ) e. CC )
64		addcl 8268	. . . . 5 |- ( ( m e. CC /\ z e. CC ) -> ( m + z ) e. CC )
65	64	adantl 277	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ ( m e. CC /\ z e. CC ) ) -> ( m + z ) e. CC )
66	33, 34, 35, 36, 57, 63, 65	seq3id2 10912	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` N ) = ( seq M ( + , F ) ` n ) )
67	66	eqcomd 2240	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` n ) = ( seq M ( + , F ) ` N ) )
68	1, 4, 6, 31, 67	climconst 12000	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   \ cdif 3211   C_ wss 3214   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   ~~> cli 11988

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-fz 10362  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-rsqrt 11708  df-abs 11709  df-clim 11989

This theorem is referenced by:  summodclem2a  12092  fsum3cvg2  12105