Theorem fsum3cvg2 11429

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 2332	. . . 4 |- F/_ m if ( k e. A , B , 0 )
2		nfv 1539	. . . . 5 |- F/ k m e. A
3		nfcsb1v 3105	. . . . 5 |- F/_ k [_ m / k ]_ B
4		nfcv 2332	. . . . 5 |- F/_ k 0
5	2, 3, 4	nfif 3577	. . . 4 |- F/_ k if ( m e. A , [_ m / k ]_ B , 0 )
6		eleq1w 2250	. . . . 5 |- ( k = m -> ( k e. A <-> m e. A ) )
7		csbeq1a 3081	. . . . 5 |- ( k = m -> B = [_ m / k ]_ B )
8	6, 7	ifbieq1d 3571	. . . 4 |- ( k = m -> if ( k e. A , B , 0 ) = if ( m e. A , [_ m / k ]_ B , 0 ) )
9	1, 5, 8	cbvmpt 4113	. . 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 2563	. . . 4 |- ( ph -> A. k e. A B e. CC )
12	3	nfel1 2343	. . . . 5 |- F/ k [_ m / k ]_ B e. CC
13	7	eleq1d 2258	. . . . 5 |- ( k = m -> ( B e. CC <-> [_ m / k ]_ B e. CC ) )
14	12, 13	rspc 2850	. . . 4 |- ( m e. A -> ( A. k e. A B e. CC -> [_ m / k ]_ B e. CC ) )
15	11, 14	mpan9 281	. . 3 |- ( ( ph /\ m e. A ) -> [_ m / k ]_ B e. CC )
16	6	dcbid 839	. . . 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 2563	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M )DECID k e. A )
19	18	adantr 276	. . . 4 |- ( ( ph /\ m e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M )DECID k e. A )
20		simpr 110	. . . 4 |- ( ( ph /\ m e. ( ZZ>= ` M ) ) -> m e. ( ZZ>= ` M ) )
21	16, 19, 20	rspcdva 2861	. . 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 11413	. 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 9558	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
26	22, 25	syl 14	. . 3 |- ( ph -> M e. ZZ )
27		fveq2 5531	. . . . 5 |- ( k = x -> ( F ` k ) = ( F ` x ) )
28	27	eleq1d 2258	. . . 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 477	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
31		0cnd 7975	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. A ) -> 0 e. CC )
32	30, 31, 17	ifcldadc 3578	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
33	29, 32	eqeltrd 2266	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	33	ralrimiva 2563	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
35	34	adantr 276	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
36		simpr 110	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
37	28, 35, 36	rspcdva 2861	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
38		eluzelz 9562	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
39		eqid 2189	. . . . . . . 8 |- ( k e. ZZ |-> if ( k e. A , B , 0 ) ) = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
40	39	fvmpt2 5616	. . . . . . 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 594	. . . . . 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 2225	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
43	42	ralrimiva 2563	. . . 4 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
44		nffvmpt1 5542	. . . . . 6 |- F/_ k ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
45	44	nfeq2 2344	. . . . 5 |- F/ k ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
46		fveq2 5531	. . . . . 6 |- ( k = n -> ( F ` k ) = ( F ` n ) )
47		fveq2 5531	. . . . . 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 2204	. . . . 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 2850	. . . 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 281	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n ) )
51		addcl 7961	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
52	51	adantl 277	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
53	26, 37, 50, 52	seq3feq 10498	. 2 |- ( ph -> seq M ( + , F ) = seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) )
54	53	fveq1d 5533	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) = ( seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ` N ) )
55	24, 53, 54	3brtr4d 4050	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   e. wcel 2160   A.wral 2468   [_csb 3072   C_ wss 3144   ifcif 3549   class class class wbr 4018   |-> cmpt 4079   ` cfv 5232  (class class class)co 5892   CCcc 7834   0cc0 7836   + caddc 7839   ZZcz 9278   ZZ>=cuz 9553   ...cfz 10033   seqcseq 10471   ~~> cli 11313

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-n0 9202  df-z 9279  df-uz 9554  df-rp 9679  df-fz 10034  df-seqfrec 10472  df-exp 10546  df-cj 10878  df-rsqrt 11034  df-abs 11035  df-clim 11314

This theorem is referenced by:  fsumsersdc  11430  fsum3cvg3  11431  ef0lem  11695