Theorem fsum3cvg 11385

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, 12-Nov-2022.)

Hypotheses

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

Assertion

Ref	Expression
fsum3cvg	|- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

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

Allowed substitution hint:   B( k)

Proof of Theorem fsum3cvg

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

Step	Hyp	Ref	Expression
1		eqid 2177	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		isumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9536	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. 2 |- ( ph -> N e. ZZ )
5		seqex 10446	. . 3 |- seq M ( + , F ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( + , F ) e. _V )
7		eqid 2177	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
8		eluzel2 9532	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9	2, 8	syl 14	. . . 4 |- ( ph -> M e. ZZ )
10		eluzelz 9536	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
11	10	adantl 277	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ZZ )
12		iftrue 3539	. . . . . . . . . . 11 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
13	12	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) = B )
14		isummo.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. CC )
15	13, 14	eqeltrd 2254	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
16	15	ex 115	. . . . . . . 8 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
17	16	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
18		iffalse 3542	. . . . . . . . 9 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
19		0cn 7948	. . . . . . . . 9 |- 0 e. CC
20	18, 19	eqeltrdi 2268	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
21	20	a1i 9	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( -. k e. A -> if ( k e. A , B , 0 ) e. CC ) )
22		isummo.dc	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
23		exmiddc 836	. . . . . . . 8 |- (DECID k e. A -> ( k e. A \/ -. k e. A ) )
24	22, 23	syl 14	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( k e. A \/ -. k e. A ) )
25	17, 21, 24	mpjaod 718	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , B , 0 ) e. CC )
26		isummo.1	. . . . . . 7 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
27	26	fvmpt2 5599	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 0 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
28	11, 25, 27	syl2anc 411	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
29	28, 25	eqeltrd 2254	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	7, 9, 29	serf 10473	. . 3 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> CC )
31	30, 2	ffvelcdmd 5652	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
32		addid1 8094	. . . . 5 |- ( m e. CC -> ( m + 0 ) = m )
33	32	adantl 277	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. CC ) -> ( m + 0 ) = m )
34	2	adantr 276	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
35		simpr 110	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> n e. ( ZZ>= ` N ) )
36	31	adantr 276	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` N ) e. CC )
37		elfzuz 10020	. . . . . 6 |- ( m e. ( ( N + 1 ) ... n ) -> m e. ( ZZ>= ` ( N + 1 ) ) )
38		eluzelz 9536	. . . . . . . . 9 |- ( m e. ( ZZ>= ` ( N + 1 ) ) -> m e. ZZ )
39	38	adantl 277	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ZZ )
40		fisumcvg.4	. . . . . . . . . . . 12 |- ( ph -> A C_ ( M ... N ) )
41	40	sseld 3154	. . . . . . . . . . 11 |- ( ph -> ( m e. A -> m e. ( M ... N ) ) )
42		fznuz 10101	. . . . . . . . . . 11 |- ( m e. ( M ... N ) -> -. m e. ( ZZ>= ` ( N + 1 ) ) )
43	41, 42	syl6 33	. . . . . . . . . 10 |- ( ph -> ( m e. A -> -. m e. ( ZZ>= ` ( N + 1 ) ) ) )
44	43	con2d 624	. . . . . . . . 9 |- ( ph -> ( m e. ( ZZ>= ` ( N + 1 ) ) -> -. m e. A ) )
45	44	imp 124	. . . . . . . 8 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> -. m e. A )
46	39, 45	eldifd 3139	. . . . . . 7 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> m e. ( ZZ \ A ) )
47		fveqeq2 5524	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) = 0 <-> ( F ` m ) = 0 ) )
48		eldifi 3257	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
49		eldifn 3258	. . . . . . . . . . . 12 |- ( k e. ( ZZ \ A ) -> -. k e. A )
50	49, 18	syl 14	. . . . . . . . . . 11 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
51	50, 19	eqeltrdi 2268	. . . . . . . . . 10 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
52	48, 51, 27	syl2anc 411	. . . . . . . . 9 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
53	52, 50	eqtrd 2210	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
54	47, 53	vtoclga 2803	. . . . . . 7 |- ( m e. ( ZZ \ A ) -> ( F ` m ) = 0 )
55	46, 54	syl 14	. . . . . 6 |- ( ( ph /\ m e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` m ) = 0 )
56	37, 55	sylan2 286	. . . . 5 |- ( ( ph /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
57	56	adantlr 477	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ( N + 1 ) ... n ) ) -> ( F ` m ) = 0 )
58		fveq2 5515	. . . . . 6 |- ( k = m -> ( F ` k ) = ( F ` m ) )
59	58	eleq1d 2246	. . . . 5 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
60	29	ralrimiva 2550	. . . . . 6 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
61	60	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
62		simpr 110	. . . . 5 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> m e. ( ZZ>= ` M ) )
63	59, 61, 62	rspcdva 2846	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ m e. ( ZZ>= ` M ) ) -> ( F ` m ) e. CC )
64		addcl 7935	. . . . 5 |- ( ( m e. CC /\ z e. CC ) -> ( m + z ) e. CC )
65	64	adantl 277	. . . 4 |- ( ( ( ph /\ n e. ( ZZ>= ` N ) ) /\ ( m e. CC /\ z e. CC ) ) -> ( m + z ) e. CC )
66	33, 34, 35, 36, 57, 63, 65	seq3id2 10508	. . 3 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` N ) = ( seq M ( + , F ) ` n ) )
67	66	eqcomd 2183	. 2 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( seq M ( + , F ) ` n ) = ( seq M ( + , F ) ` N ) )
68	1, 4, 6, 31, 67	climconst 11297	1 |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   \ cdif 3126   C_ wss 3129   ifcif 3534   class class class wbr 4003   |-> cmpt 4064   ` cfv 5216  (class class class)co 5874   CCcc 7808   0cc0 7810   1c1 7811   + caddc 7813   ZZcz 9252   ZZ>=cuz 9527   ...cfz 10007   seqcseq 10444   ~~> cli 11285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-n0 9176  df-z 9253  df-uz 9528  df-rp 9653  df-fz 10008  df-seqfrec 10445  df-exp 10519  df-cj 10850  df-rsqrt 11006  df-abs 11007  df-clim 11286

This theorem is referenced by:  summodclem2a  11388  fsum3cvg2  11401