Theorem fisumcvg 10572

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

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
fisumcvg	|- ( ph -> seq M ( + , F , CC ) ~~> ( seq M ( + , F , CC ) ` N ) )

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

Proof of Theorem fisumcvg

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

Step	Hyp	Ref	Expression
1		eqid 2083	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		isumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 8921	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. 2 |- ( ph -> N e. ZZ )
5		iseqex 9740	. . 3 |- seq M ( + , F , CC ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( + , F , CC ) e. _V )
7		eqid 2083	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 8917	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 14	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 8921	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 271	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 3378	. . . . . . . . . . 11 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
13	12	adantl 271	. . . . . . . . . 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 2159	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
16	15	ex 113	. . . . . . . 8 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
17	16	adantr 270	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
18		iffalse 3381	. . . . . . . . 9 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
19		0cn 7381	. . . . . . . . 9 |- 0 e. CC
20	18, 19	syl6eqel 2173	. . . . . . . 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 778	. . . . . . . 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 671	. . . . . 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 5329	. . . . . 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 403	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
29	28, 25	eqeltrd 2159	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	7, 9, 29	iserf 9766	. . 3 |- ( ph -> seq M ( + , F , CC ) : ( ZZ>= ` M ) --> CC )
31	30, 2	ffvelrnd 5378	. 2 |- ( ph -> ( seq M ( + , F , CC ) ` N ) e. CC )
32		addid1 7521	. . . . 5 |- ( m e. CC -> ( m + 0 ) = m )
33	32	adantl 271	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. CC ) -> ( m + 0 ) = m )
34	2	adantr 270	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
35		simpr 108	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> n e. ( ZZ>= ` N ) )
36	31	adantr 270	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F , CC ) ` N ) e. CC )
37		elfzuz 9329	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
38		eluzelz 8921	. . . . . . . . 9 |- ( m e. ( ZZ>= ` ( N + 1 ) ) -> m e. ZZ )
39	38	adantl 271	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ZZ )
40		fisumcvg.4	. . . . . . . . . . . 12 |- ( ph -> A C_ ( M ... N ) )
41	40	sseld 3009	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
42		fznuz 9407	. . . . . . . . . . 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 587	. . . . . . . . 9 |- ( ph -> ( m e. ( ZZ>= ` ( N + 1 ) ) -> -. m e. A ) )
45	44	imp 122	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> -. m e. A )
46	39, 45	eldifd 2994	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
47		fveq2 5251	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
48	47	eqeq1d 2091	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 0 <-> ( F ` m ) = 0 ) )
49		eldifi 3106	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
50		eldifn 3107	. . . . . . . . . . . 12 |- ( k e. ( ZZ \ A ) -> -. k e. A )
51	50, 18	syl 14	. . . . . . . . . . 11 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
52	51, 19	syl6eqel 2173	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
53	49, 52, 27	syl2anc 403	. . . . . . . . 9 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
54	53, 51	eqtrd 2115	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
55	48, 54	vtoclga 2675	. . . . . . 7 |- ( m e. ( ZZ \ A ) -> ( F ` m ) = 0 )
56	46, 55	syl 14	. . . . . 6 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` m ) = 0 )
57	37, 56	sylan2 280	. . . . 5 |- ( ( ph /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
58	57	adantlr 461	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
59		cnex 7367	. . . . 5 |- CC e. _V
60	59	a1i 9	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> CC e. _V )
61	47	eleq1d 2151	. . . . 5 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
62	29	ralrimiva 2440	. . . . . 6 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
63	62	ad2antrr 472	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
64		simpr 108	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> m e. ( ZZ>= ` M ) )
65	61, 63, 64	rspcdva 2717	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> ( F ` m ) e. CC )
66		addcl 7368	. . . . 5 |- ( ( m e. CC /\ z e. CC ) -> ( m + z ) e. CC )
67	66	adantl 271	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ ( m e. CC /\ z e. CC ) ) -> ( m + z ) e. CC )
68	33, 34, 35, 36, 58, 60, 65, 67	iseqid2 9781	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F , CC ) ` N ) = ( seq M ( + , F , CC ) ` n ) )
69	68	eqcomd 2088	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F , CC ) ` n ) = ( seq M ( + , F , CC ) ` N ) )
70	1, 4, 6, 31, 69	climconst 10501	1 |- ( ph -> seq M ( + , F , CC ) ~~> ( seq M ( + , F , CC ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 662  DECID wdc 776   = wceq 1285   e. wcel 1434   A.wral 2353   _Vcvv 2612   \ cdif 2981   C_ wss 2984   ifcif 3373   class class class wbr 3811   |-> cmpt 3865   ` cfv 4967  (class class class)co 5589   CCcc 7249   0cc0 7251   1c1 7252   + caddc 7254   ZZcz 8644   ZZ>=cuz 8912   ...cfz 9317   seqcseq 9738   ~~> cli 10489

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-mulrcl 7345  ax-addcom 7346  ax-mulcom 7347  ax-addass 7348  ax-mulass 7349  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-1rid 7353  ax-0id 7354  ax-rnegex 7355  ax-precex 7356  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-apti 7361  ax-pre-ltadd 7362  ax-pre-mulgt0 7363  ax-pre-mulext 7364

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-reap 7950  df-ap 7957  df-div 8036  df-inn 8315  df-2 8373  df-n0 8564  df-z 8645  df-uz 8913  df-rp 9028  df-fz 9318  df-iseq 9739  df-iexp 9790  df-cj 10101  df-rsqrt 10256  df-abs 10257  df-clim 10490

This theorem is referenced by: (None)