Theorem fsum3cvg2 11537

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 2336	. . . 4 |- F/_ m if ( k e. A , B , 0 )
2		nfv 1539	. . . . 5 |- F/ k m e. A
3		nfcsb1v 3113	. . . . 5 |- F/_ k [_ m / k ]_ B
4		nfcv 2336	. . . . 5 |- F/_ k 0
5	2, 3, 4	nfif 3585	. . . 4 |- F/_ k if ( m e. A , [_ m / k ]_ B , 0 )
6		eleq1w 2254	. . . . 5 |- ( k = m -> ( k e. A <-> m e. A ) )
7		csbeq1a 3089	. . . . 5 |- ( k = m -> B = [_ m / k ]_ B )
8	6, 7	ifbieq1d 3579	. . . 4 |- ( k = m -> if ( k e. A , B , 0 ) = if ( m e. A , [_ m / k ]_ B , 0 ) )
9	1, 5, 8	cbvmpt 4124	. . 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 2567	. . . 4 |- ( ph -> A. k e. A B e. CC )
12	3	nfel1 2347	. . . . 5 |- F/ k [_ m / k ]_ B e. CC
13	7	eleq1d 2262	. . . . 5 |- ( k = m -> ( B e. CC <-> [_ m / k ]_ B e. CC ) )
14	12, 13	rspc 2858	. . . 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 2567	. . . . 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 2869	. . 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 11521	. 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 9597	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
26	22, 25	syl 14	. . 3 |- ( ph -> M e. ZZ )
27		fveq2 5554	. . . . 5 |- ( k = x -> ( F ` k ) = ( F ` x ) )
28	27	eleq1d 2262	. . . 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 8012	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. A ) -> 0 e. CC )
32	30, 31, 17	ifcldadc 3586	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
33	29, 32	eqeltrd 2270	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	33	ralrimiva 2567	. . . . 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 2869	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
38		eluzelz 9601	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
39		eqid 2193	. . . . . . . 8 |- ( k e. ZZ |-> if ( k e. A , B , 0 ) ) = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
40	39	fvmpt2 5641	. . . . . . 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 2229	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
43	42	ralrimiva 2567	. . . 4 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
44		nffvmpt1 5565	. . . . . 6 |- F/_ k ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
45	44	nfeq2 2348	. . . . 5 |- F/ k ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
46		fveq2 5554	. . . . . 6 |- ( k = n -> ( F ` k ) = ( F ` n ) )
47		fveq2 5554	. . . . . 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 2208	. . . . 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 2858	. . . 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 7997	. . . 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 10551	. 2 |- ( ph -> seq M ( + , F ) = seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) )
54	53	fveq1d 5556	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) = ( seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ` N ) )
55	24, 53, 54	3brtr4d 4061	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 835   = wceq 1364   e. wcel 2164   A.wral 2472   [_csb 3080   C_ wss 3153   ifcif 3557   class class class wbr 4029   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   + 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:  fsumsersdc  11538  fsum3cvg3  11539  ef0lem  11803