Theorem isumclim3 11224

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 11096	. . 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 2141	. . . 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 5583	. . . . 5 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
9	8	ffvelrnda 5563	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
10	4, 5, 6, 9	isum 11186	. . 3 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
11	7	ralrimiva 2508	. . . 4 |- ( ph -> A. k e. Z A e. CC )
12		sumfct 11175	. . . 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 10251	. . . . . . 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		fzssuz 9876	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
19	18, 4	sseqtrri 3137	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
20		resmpt 4875	. . . . . . . . . . . . 13 |- ( ( M ... j ) C_ Z -> ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A ) )
21	19, 20	ax-mp 5	. . . . . . . . . . . 12 |- ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A )
22	21	fveq1i 5430	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
23		fvres 5453	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
24	22, 23	syl5reqr 2188	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
25	24	sumeq2i 11165	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
26		ssralv 3166	. . . . . . . . . . 11 |- ( ( M ... j ) C_ Z -> ( A. k e. Z A e. CC -> A. k e. ( M ... j ) A e. CC ) )
27	19, 11, 26	mpsyl 65	. . . . . . . . . 10 |- ( ph -> A. k e. ( M ... j ) A e. CC )
28		sumfct 11175	. . . . . . . . . 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 2185	. . . . . . . 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 2141	. . . . . . . 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 2233	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
35	4	eleq2i 2207	. . . . . . . . . 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 11198	. . . . . . 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 2180	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
40	4, 15, 1, 5, 39	climeq 11100	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
41	40	iotabidv 5117	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
42		df-fv 5139	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
43		df-fv 5139	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
44	41, 42, 43	3eqtr4g 2198	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
45	10, 13, 44	3eqtr3d 2181	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
46	3, 45	breqtrrd 3964	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   A.wral 2417   _Vcvv 2689   C_ wss 3076   class class class wbr 3937   |-> cmpt 3997   dom cdm 4547   |` cres 4549   iotacio 5094   ` cfv 5131  (class class class)co 5782   CCcc 7642   + caddc 7647   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821   seqcseq 10249   ~~> cli 11079   sum_csu 11154

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by: (None)