Theorem isumclim3 11364

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 11236	. . 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 2166	. . . 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 5640	. . . . 5 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
9	8	ffvelrnda 5620	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
10	4, 5, 6, 9	isum 11326	. . 3 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
11	7	ralrimiva 2539	. . . 4 |- ( ph -> A. k e. Z A e. CC )
12		sumfct 11315	. . . 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 10382	. . . . . . 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 5510	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
19		fzssuz 10000	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
20	19, 4	sseqtrri 3177	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
21		resmpt 4932	. . . . . . . . . . . . 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 5487	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
24	18, 23	eqtr3di 2214	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
25	24	sumeq2i 11305	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
26		ssralv 3206	. . . . . . . . . . 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 11315	. . . . . . . . . 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	syl5eq 2211	. . . . . . . 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 2166	. . . . . . . 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 2259	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
35	4	eleq2i 2233	. . . . . . . . . 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 11338	. . . . . . 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 2205	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
40	4, 15, 1, 5, 39	climeq 11240	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
41	40	iotabidv 5174	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
42		df-fv 5196	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
43		df-fv 5196	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
44	41, 42, 43	3eqtr4g 2224	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
45	10, 13, 44	3eqtr3d 2206	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
46	3, 45	breqtrrd 4010	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   class class class wbr 3982   |-> cmpt 4043   dom cdm 4604   |` cres 4606   iotacio 5151   ` cfv 5188  (class class class)co 5842   CCcc 7751   + caddc 7756   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   seqcseq 10380   ~~> cli 11219   sum_csu 11294

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by: (None)