Theorem fsum3cvg2 11291

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 2299	. . . 4 |- F/_ m if ( k e. A , B , 0 )
2		nfv 1508	. . . . 5 |- F/ k m e. A
3		nfcsb1v 3064	. . . . 5 |- F/_ k [_ m / k ]_ B
4		nfcv 2299	. . . . 5 |- F/_ k 0
5	2, 3, 4	nfif 3533	. . . 4 |- F/_ k if ( m e. A , [_ m / k ]_ B , 0 )
6		eleq1w 2218	. . . . 5 |- ( k = m -> ( k e. A <-> m e. A ) )
7		csbeq1a 3040	. . . . 5 |- ( k = m -> B = [_ m / k ]_ B )
8	6, 7	ifbieq1d 3527	. . . 4 |- ( k = m -> if ( k e. A , B , 0 ) = if ( m e. A , [_ m / k ]_ B , 0 ) )
9	1, 5, 8	cbvmpt 4059	. . 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 2530	. . . 4 |- ( ph -> A. k e. A B e. CC )
12	3	nfel1 2310	. . . . 5 |- F/ k [_ m / k ]_ B e. CC
13	7	eleq1d 2226	. . . . 5 |- ( k = m -> ( B e. CC <-> [_ m / k ]_ B e. CC ) )
14	12, 13	rspc 2810	. . . 4 |- ( m e. A -> ( A. k e. A B e. CC -> [_ m / k ]_ B e. CC ) )
15	11, 14	mpan9 279	. . 3 |- ( ( ph /\ m e. A ) -> [_ m / k ]_ B e. CC )
16	6	dcbid 824	. . . 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 2530	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M )DECID k e. A )
19	18	adantr 274	. . . 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 2821	. . 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 11275	. 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 9444	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
26	22, 25	syl 14	. . 3 |- ( ph -> M e. ZZ )
27		fveq2 5468	. . . . 5 |- ( k = x -> ( F ` k ) = ( F ` x ) )
28	27	eleq1d 2226	. . . 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 469	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. A ) -> B e. CC )
31		0cnd 7871	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. A ) -> 0 e. CC )
32	30, 31, 17	ifcldadc 3534	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
33	29, 32	eqeltrd 2234	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	33	ralrimiva 2530	. . . . 5 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
35	34	adantr 274	. . . 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 2821	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
38		eluzelz 9448	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
39		eqid 2157	. . . . . . . 8 |- ( k e. ZZ |-> if ( k e. A , B , 0 ) ) = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
40	39	fvmpt2 5551	. . . . . . 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 584	. . . . . 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 2193	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
43	42	ralrimiva 2530	. . . 4 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
44		nffvmpt1 5479	. . . . . 6 |- F/_ k ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
45	44	nfeq2 2311	. . . . 5 |- F/ k ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
46		fveq2 5468	. . . . . 6 |- ( k = n -> ( F ` k ) = ( F ` n ) )
47		fveq2 5468	. . . . . 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 2172	. . . . 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 2810	. . . 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 279	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n ) )
51		addcl 7857	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
52	51	adantl 275	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
53	26, 37, 50, 52	seq3feq 10371	. 2 |- ( ph -> seq M ( + , F ) = seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) )
54	53	fveq1d 5470	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) = ( seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ` N ) )
55	24, 53, 54	3brtr4d 3996	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103  DECID wdc 820   = wceq 1335   e. wcel 2128   A.wral 2435   [_csb 3031   C_ wss 3102   ifcif 3505   class class class wbr 3965   |-> cmpt 4025   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732   + caddc 7735   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912   seqcseq 10344   ~~> cli 11175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-fz 9913  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-rsqrt 10898  df-abs 10899  df-clim 11176

This theorem is referenced by:  fsumsersdc  11292  fsum3cvg3  11293  ef0lem  11557