Theorem fsum3cvg2 11705

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 2348	. . . 4 |- F/_ m if ( k e. A , B , 0 )
2		nfv 1551	. . . . 5 |- F/ k m e. A
3		nfcsb1v 3126	. . . . 5 |- F/_ k [_ m / k ]_ B
4		nfcv 2348	. . . . 5 |- F/_ k 0
5	2, 3, 4	nfif 3599	. . . 4 |- F/_ k if ( m e. A , [_ m / k ]_ B , 0 )
6		eleq1w 2266	. . . . 5 |- ( k = m -> ( k e. A <-> m e. A ) )
7		csbeq1a 3102	. . . . 5 |- ( k = m -> B = [_ m / k ]_ B )
8	6, 7	ifbieq1d 3593	. . . 4 |- ( k = m -> if ( k e. A , B , 0 ) = if ( m e. A , [_ m / k ]_ B , 0 ) )
9	1, 5, 8	cbvmpt 4139	. . 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 2579	. . . 4 |- ( ph -> A. k e. A B e. CC )
12	3	nfel1 2359	. . . . 5 |- F/ k [_ m / k ]_ B e. CC
13	7	eleq1d 2274	. . . . 5 |- ( k = m -> ( B e. CC <-> [_ m / k ]_ B e. CC ) )
14	12, 13	rspc 2871	. . . 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 840	. . . 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 2579	. . . . 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 2882	. . 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 11689	. 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 9653	. . . 4 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
26	22, 25	syl 14	. . 3 |- ( ph -> M e. ZZ )
27		fveq2 5576	. . . . 5 |- ( k = x -> ( F ` k ) = ( F ` x ) )
28	27	eleq1d 2274	. . . 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 8065	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. A ) -> 0 e. CC )
32	30, 31, 17	ifcldadc 3600	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
33	29, 32	eqeltrd 2282	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	33	ralrimiva 2579	. . . . 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 2882	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
38		eluzelz 9657	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
39		eqid 2205	. . . . . . . 8 |- ( k e. ZZ |-> if ( k e. A , B , 0 ) ) = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
40	39	fvmpt2 5663	. . . . . . 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 2241	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
43	42	ralrimiva 2579	. . . 4 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` k ) )
44		nffvmpt1 5587	. . . . . 6 |- F/_ k ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
45	44	nfeq2 2360	. . . . 5 |- F/ k ( F ` n ) = ( ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ` n )
46		fveq2 5576	. . . . . 6 |- ( k = n -> ( F ` k ) = ( F ` n ) )
47		fveq2 5576	. . . . . 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 2220	. . . . 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 2871	. . . 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 8050	. . . 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 10625	. 2 |- ( ph -> seq M ( + , F ) = seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) )
54	53	fveq1d 5578	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) = ( seq M ( + , ( k e. ZZ |-> if ( k e. A , B , 0 ) ) ) ` N ) )
55	24, 53, 54	3brtr4d 4076	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 836   = wceq 1373   e. wcel 2176   A.wral 2484   [_csb 3093   C_ wss 3166   ifcif 3571   class class class wbr 4044   |-> cmpt 4105   ` cfv 5271  (class class class)co 5944   CCcc 7923   0cc0 7925   + caddc 7928   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   ~~> cli 11589

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-fz 10131  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-rsqrt 11309  df-abs 11310  df-clim 11590

This theorem is referenced by:  fsumsersdc  11706  fsum3cvg3  11707  ef0lem  11971