Theorem fsum3cvg2 10841

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, 2-Dec-2022.)

Hypotheses

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

Assertion

Ref	Expression
fsum3cvg2	|- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

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

Allowed substitution hint:   B( k)

Proof of Theorem fsum3cvg2

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

Step	Hyp	Ref	Expression
1		nfcv 2229	. . . 4 |- F/_ m if ( k e. A , B , 0 )
2		nfv 1467	. . . . 5 |- F/ k m e. A
3		nfcsb1v 2964	. . . . 5 |- F/_ k [_ m / k ]_ B
4		nfcv 2229	. . . . 5 |- F/_ k 0
5	2, 3, 4	nfif 3423	. . . 4 |- F/_ k if ( m e. A , [_ m / k ]_ B , 0 )
6		eleq1w 2149	. . . . 5 |- ( k = m -> ( k e. A <-> m e. A ) )
7		csbeq1a 2942	. . . . 5 |- ( k = m -> B = [_ m / k ]_ B )
8	6, 7	ifbieq1d 3417	. . . 4 |- ( k = m -> if ( k e. A , B , 0 ) = if ( m e. A , [_ m / k ]_ B , 0 ) )
9	1, 5, 8	cbvmpt 3939	. . 3 |- ( k e. ZZ |-> if ( k e. A , B , 0 ) ) = ( m e. ZZ |-> if ( m e. A , [_ m / k ]_ B , 0 ) )
10		fsumsers.3	. . . . 5 |- ( ( ph /\ k e. A ) -> B e. CC )
11	10	ralrimiva 2447	. . . 4 |- ( ph -> A. k e. A B e. CC )
12	3	nfel1 2240	. . . . 5 |- F/ k [_ m / k ]_ B e. CC
13	7	eleq1d 2157	. . . . 5 |- ( k = m -> ( B e. CC <-> [_ m / k ]_ B e. CC ) )
14	12, 13	rspc 2717	. . . 4 |- ( m e. A -> ( A. k e. A B e. CC -> [_ m / k ]_ B e. CC ) )
15	11, 14	mpan9 276	. . 3 |- ( ( ph /\ m e. A ) -> [_ m / k ]_ B e. CC )
16	6	dcbid 787	. . . 4 |- ( k = m -> (DECID k e. A <-> DECID m e. A ) )
17		fsumsers.dc	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
18	17	ralrimiva 2447	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M )DECID k e. A )
19	18	adantr 271	. . . 4 |- ( ( ph /\ m e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M )DECID k e. A )
20		simpr 109	. . . 4 |- ( ( ph /\ m e. ( ZZ>= ` M ) ) -> m e. ( ZZ>= ` M ) )
21	16, 19, 20	rspcdva 2728	. . 3 |- ( ( ph /\ m e. ( ZZ>= ` M ) ) -> DECID m e. A )
22		fsumsers.2	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
23		fsumsers.4	. . 3 |- ( ph -> A C_ ( M ... N ) )
24	9, 15, 21, 22, 23	fsum3cvg 10821	. 2 |- ( ph -> seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ~~> ( seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ` N ) )
25		eluzel2 9078	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
26	22, 25	syl 14	. . 3 |- ( ph -> M e. ZZ )
27		fveq2 5318	. . . . 5 |- ( k = x -> ( F ` k ) = ( F ` x ) )
28	27	eleq1d 2157	. . . 4 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
29		fsumsers.1	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
30	10	adantlr 462	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
31		0cnd 7535	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. A ) -> 0 e. CC )
32	30, 31, 17	ifcldadc 3424	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
33	29, 32	eqeltrd 2165	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	33	ralrimiva 2447	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
35	34	adantr 271	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
36		simpr 109	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
37	28, 35, 36	rspcdva 2728	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
38		eluzelz 9082	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
39		eqid 2089	. . . . . . . 8 |- ( k e. ZZ |-> if ( k e. A , B , 0 ) ) = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
40	39	fvmpt2 5399	. . . . . . 7 |- ( ( k e. ZZ /\ if ( k e. A , B , 0 ) e. CC ) -> ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) = if ( k e. A , B , 0 ) )
41	38, 32, 40	syl2an2 562	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) = if ( k e. A , B , 0 ) )
42	29, 41	eqtr4d 2124	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
43	42	ralrimiva 2447	. . . 4 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
44		nffvmpt1 5329	. . . . . 6 |- F/_ k ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
45	44	nfeq2 2241	. . . . 5 |- F/ k ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
46		fveq2 5318	. . . . . 6 |- ( k = n -> ( F ` k ) = ( F ` n ) )
47		fveq2 5318	. . . . . 6 |- ( k = n -> ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n ) )
48	46, 47	eqeq12d 2103	. . . . 5 |- ( k = n -> ( ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) <-> ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n ) ) )
49	45, 48	rspc 2717	. . . 4 |- ( n e. ( ZZ>= ` M ) -> ( A. k e. ( ZZ>= ` M ) ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) -> ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n ) ) )
50	43, 49	mpan9 276	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n ) )
51		addcl 7521	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
52	51	adantl 272	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
53	26, 37, 50, 52	seq3feq 9951	. 2 |- ( ph -> seq M ( + , F ) = seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) )
54	53	fveq1d 5320	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) = ( seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ` N ) )
55	24, 53, 54	3brtr4d 3881	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103  DECID wdc 781   = wceq 1290   e. wcel 1439   A.wral 2360   [_csb 2934   C_ wss 3000   ifcif 3397   class class class wbr 3851   |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7402   0cc0 7404   + caddc 7407   ZZcz 8804   ZZ>=cuz 9073   ...cfz 9478   seqcseq 9906   ~~> cli 10720

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-n0 8728  df-z 8805  df-uz 9074  df-rp 9189  df-fz 9479  df-iseq 9907  df-seq3 9908  df-exp 10009  df-cj 10330  df-rsqrt 10485  df-abs 10486  df-clim 10721

This theorem is referenced by:  fsum3cvg3  10843  ef0lem  11004