Theorem isumclim3 12168

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 12039	. . 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 2239	. . . 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 5854	. . . . 5 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
9	8	ffvelcdmda 5834	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
10	4, 5, 6, 9	isum 12130	. . 3 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
11	7	ralrimiva 2623	. . . 4 |- ( ph -> A. k e. Z A e. CC )
12		sumfct 12118	. . . 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 10864	. . . . . . 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 5714	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
19		fzssuz 10449	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
20	19, 4	sseqtrri 3283	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
21		resmpt 5106	. . . . . . . . . . . . 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 5691	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
24	18, 23	eqtr3di 2286	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
25	24	sumeq2i 12108	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
26		ssralv 3312	. . . . . . . . . . 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 12118	. . . . . . . . . 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 2283	. . . . . . . 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 2239	. . . . . . . 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 2331	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
35	4	eleq2i 2305	. . . . . . . . . 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 12142	. . . . . . 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 2278	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
40	4, 15, 1, 5, 39	climeq 12043	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
41	40	iotabidv 5355	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
42		df-fv 5380	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
43		df-fv 5380	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
44	41, 42, 43	3eqtr4g 2296	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
45	10, 13, 44	3eqtr3d 2279	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
46	3, 45	breqtrrd 4153	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   |-> cmpt 4187   dom cdm 4769   |` cres 4771   iotacio 5330   ` cfv 5372  (class class class)co 6075   CCcc 8167   + caddc 8172   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   ~~> cli 12022   sum_csu 12097

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by: (None)