Theorem isumclim3 11386

Description: The sequence of partial finite sums of a converging infinite series converges to the infinite sum of the series. Note that j must not occur in A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
isumclim3.1	|- Z = ( ZZ>= ` M )
isumclim3.2	|- ( ph -> M e. ZZ )
isumclim3.3	|- ( ph -> F e. dom ~~> )
isumclim3.4	|- ( ( ph /\ k e. Z ) -> A e. CC )
isumclim3.5	|- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )

Assertion

Ref	Expression
isumclim3	|- ( ph -> F ~~> sum_ k e. Z A )

Distinct variable groups:   A, j   j, k, M   ph, j, k   j, Z, k   j, F

Allowed substitution hints:   A( k)   F( k)

Proof of Theorem isumclim3

Dummy variables m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumclim3.3	. . 3 |- ( ph -> F e. dom ~~> )
2		climdm 11258	. . 3 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
3	1, 2	sylib 121	. 2 |- ( ph -> F ~~> ( ~~> ` F ) )
4		isumclim3.1	. . . 4 |- Z = ( ZZ>= ` M )
5		isumclim3.2	. . . 4 |- ( ph -> M e. ZZ )
6		eqidd 2171	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
7		isumclim3.4	. . . . . 6 |- ( ( ph /\ k e. Z ) -> A e. CC )
8	7	fmpttd 5651	. . . . 5 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
9	8	ffvelrnda 5631	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
10	4, 5, 6, 9	isum 11348	. . 3 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
11	7	ralrimiva 2543	. . . 4 |- ( ph -> A. k e. Z A e. CC )
12		sumfct 11337	. . . 4 |- ( A. k e. Z A e. CC -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = sum_ k e. Z A )
13	11, 12	syl 14	. . 3 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = sum_ k e. Z A )
14		seqex 10403	. . . . . . 7 |- seq M ( + , ( k e. Z |-> A ) ) e. _V
15	14	a1i 9	. . . . . 6 |- ( ph -> seq M ( + , ( k e. Z |-> A ) ) e. _V )
16		isumclim3.5	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )
17		simpl 108	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> ph )
18		fvres 5520	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
19		fzssuz 10021	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
20	19, 4	sseqtrri 3182	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
21		resmpt 4939	. . . . . . . . . . . . 13 |- ( ( M ... j ) C_ Z -> ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A ) )
22	20, 21	ax-mp 5	. . . . . . . . . . . 12 |- ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A )
23	22	fveq1i 5497	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
24	18, 23	eqtr3di 2218	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
25	24	sumeq2i 11327	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
26		ssralv 3211	. . . . . . . . . . 11 |- ( ( M ... j ) C_ Z -> ( A. k e. Z A e. CC -> A. k e. ( M ... j ) A e. CC ) )
27	20, 11, 26	mpsyl 65	. . . . . . . . . 10 |- ( ph -> A. k e. ( M ... j ) A e. CC )
28		sumfct 11337	. . . . . . . . . 10 |- ( A. k e. ( M ... j ) A e. CC -> sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m ) = sum_ k e. ( M ... j ) A )
29	27, 28	syl 14	. . . . . . . . 9 |- ( ph -> sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m ) = sum_ k e. ( M ... j ) A )
30	25, 29	eqtrid 2215	. . . . . . . 8 |- ( ph -> sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ k e. ( M ... j ) A )
31	17, 30	syl 14	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ k e. ( M ... j ) A )
32		eqidd 2171	. . . . . . . 8 |- ( ( ( ph /\ j e. Z ) /\ m e. ( ZZ>= ` M ) ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
33		simpr 109	. . . . . . . . 9 |- ( ( ph /\ j e. Z ) -> j e. Z )
34	33, 4	eleqtrdi 2263	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
35	4	eleq2i 2237	. . . . . . . . . 10 |- ( m e. Z <-> m e. ( ZZ>= ` M ) )
36	35	biimpri 132	. . . . . . . . 9 |- ( m e. ( ZZ>= ` M ) -> m e. Z )
37	17, 36, 9	syl2an 287	. . . . . . . 8 |- ( ( ( ph /\ j e. Z ) /\ m e. ( ZZ>= ` M ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
38	32, 34, 37	fsum3ser 11360	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = ( seq M ( + , ( k e. Z |-> A ) ) ` j ) )
39	16, 31, 38	3eqtr2rd 2210	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
40	4, 15, 1, 5, 39	climeq 11262	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
41	40	iotabidv 5181	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
42		df-fv 5206	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
43		df-fv 5206	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
44	41, 42, 43	3eqtr4g 2228	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
45	10, 13, 44	3eqtr3d 2211	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
46	3, 45	breqtrrd 4017	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   C_ wss 3121   class class class wbr 3989   |-> cmpt 4050   dom cdm 4611   |` cres 4613   iotacio 5158   ` cfv 5198  (class class class)co 5853   CCcc 7772   + caddc 7777   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401   ~~> cli 11241   sum_csu 11316

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by: (None)