Theorem isumclim3 11588

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 11460	. . 3 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
3	1, 2	sylib 122	. 2 |- ( ph -> F ~~> ( ~~> ` F ) )
4		isumclim3.1	. . . 4 |- Z = ( ZZ>= ` M )
5		isumclim3.2	. . . 4 |- ( ph -> M e. ZZ )
6		eqidd 2197	. . . 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 5717	. . . . 5 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
9	8	ffvelcdmda 5697	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
10	4, 5, 6, 9	isum 11550	. . 3 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
11	7	ralrimiva 2570	. . . 4 |- ( ph -> A. k e. Z A e. CC )
12		sumfct 11539	. . . 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 10541	. . . . . . 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 109	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> ph )
18		fvres 5582	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
19		fzssuz 10140	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
20	19, 4	sseqtrri 3218	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
21		resmpt 4994	. . . . . . . . . . . . 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 5559	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
24	18, 23	eqtr3di 2244	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
25	24	sumeq2i 11529	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
26		ssralv 3247	. . . . . . . . . . 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 11539	. . . . . . . . . 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 2241	. . . . . . . 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 2197	. . . . . . . 8 |- ( ( ( ph /\ j e. Z ) /\ m e. ( ZZ>= ` M ) ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
33		simpr 110	. . . . . . . . 9 |- ( ( ph /\ j e. Z ) -> j e. Z )
34	33, 4	eleqtrdi 2289	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
35	4	eleq2i 2263	. . . . . . . . . 10 |- ( m e. Z <-> m e. ( ZZ>= ` M ) )
36	35	biimpri 133	. . . . . . . . 9 |- ( m e. ( ZZ>= ` M ) -> m e. Z )
37	17, 36, 9	syl2an 289	. . . . . . . 8 |- ( ( ( ph /\ j e. Z ) /\ m e. ( ZZ>= ` M ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
38	32, 34, 37	fsum3ser 11562	. . . . . . 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 2236	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
40	4, 15, 1, 5, 39	climeq 11464	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
41	40	iotabidv 5241	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
42		df-fv 5266	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
43		df-fv 5266	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
44	41, 42, 43	3eqtr4g 2254	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
45	10, 13, 44	3eqtr3d 2237	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
46	3, 45	breqtrrd 4061	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   C_ wss 3157   class class class wbr 4033   |-> cmpt 4094   dom cdm 4663   |` cres 4665   iotacio 5217   ` cfv 5258  (class class class)co 5922   CCcc 7877   + caddc 7882   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539   ~~> cli 11443   sum_csu 11518

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519

This theorem is referenced by: (None)