Theorem fsum3cvg 11521

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 2193	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		isumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9601	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. 2 |- ( ph -> N e. ZZ )
5		seqex 10520	. . 3 |- seq M ( + , F ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( + , F ) e. _V )
7		eqid 2193	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 9597	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 14	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 9601	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 277	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 3562	. . . . . . . . . . 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 2270	. . . . . . . . 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 3565	. . . . . . . . 9 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
19		0cn 8011	. . . . . . . . 9 |- 0 e. CC
20	18, 19	eqeltrdi 2284	. . . . . . . 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 837	. . . . . . . 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 719	. . . . . 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 5641	. . . . . 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 2270	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	7, 9, 29	serf 10554	. . 3 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> CC )
31	30, 2	ffvelcdmd 5694	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
32		addrid 8157	. . . . 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 10087	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
38		eluzelz 9601	. . . . . . . . 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 3178	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
42		fznuz 10168	. . . . . . . . . . 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 625	. . . . . . . . 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 3163	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
47		fveqeq2 5563	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 0 <-> ( F ` m ) = 0 ) )
48		eldifi 3281	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
49		eldifn 3282	. . . . . . . . . . . 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 2284	. . . . . . . . . 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 2226	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
54	47, 53	vtoclga 2826	. . . . . . 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 5554	. . . . . 6 |- ( k = m -> ( F ` k ) = ( F ` m ) )
59	58	eleq1d 2262	. . . . 5 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
60	29	ralrimiva 2567	. . . . . 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 2869	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> ( F ` m ) e. CC )
64		addcl 7997	. . . . 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 10597	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` N ) = ( seq M ( + , F ) ` n ) )
67	66	eqcomd 2199	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` n ) = ( seq M ( + , F ) ` N ) )
68	1, 4, 6, 31, 67	climconst 11433	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   \ cdif 3150   C_ wss 3153   ifcif 3557   class class class wbr 4029   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   ~~> cli 11421

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-fz 10075  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-rsqrt 11142  df-abs 11143  df-clim 11422

This theorem is referenced by:  summodclem2a  11524  fsum3cvg2  11537