Theorem isummulc2 10883

Description: An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
isumcl.1	|- Z = ( ZZ>= ` M )
isumcl.2	|- ( ph -> M e. ZZ )
isumcl.3	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
isumcl.4	|- ( ( ph /\ k e. Z ) -> A e. CC )
isumcl.5	|- ( ph -> seq M ( + , F ) e. dom ~~> )
summulc.6	|- ( ph -> B e. CC )

Assertion

Ref	Expression
isummulc2	|- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )

Distinct variable groups:   B, k   k, F   ph, k   k, Z   k, M

Allowed substitution hint:   A( k)

Proof of Theorem isummulc2

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

Step	Hyp	Ref	Expression
1		isumcl.1	. . 3 |- Z = ( ZZ>= ` M )
2		isumcl.2	. . 3 |- ( ph -> M e. ZZ )
3		eqidd 2090	. . 3 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( ( k e. Z |-> ( B x. A ) ) ` m ) )
4		summulc.6	. . . . . . 7 |- ( ph -> B e. CC )
5	4	adantr 271	. . . . . 6 |- ( ( ph /\ k e. Z ) -> B e. CC )
6		isumcl.4	. . . . . 6 |- ( ( ph /\ k e. Z ) -> A e. CC )
7	5, 6	mulcld 7571	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( B x. A ) e. CC )
8	7	fmpttd 5469	. . . 4 |- ( ph -> ( k e. Z |-> ( B x. A ) ) : Z --> CC )
9	8	ffvelrnda 5450	. . 3 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) e. CC )
10	1	eleq2i 2155	. . . . . . . . 9 |- ( x e. Z <-> x e. ( ZZ>= ` M ) )
11	10	biimpri 132	. . . . . . . 8 |- ( x e. ( ZZ>= ` M ) -> x e. Z )
12	11	adantl 272	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. Z )
13	4	adantr 271	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> B e. CC )
14	6	ralrimiva 2447	. . . . . . . . . 10 |- ( ph -> A. k e. Z A e. CC )
15	14	adantr 271	. . . . . . . . 9 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. Z A e. CC )
16		nfcsb1v 2966	. . . . . . . . . . 11 |- F/_ k [_ x / k ]_ A
17	16	nfel1 2240	. . . . . . . . . 10 |- F/ k [_ x / k ]_ A e. CC
18		csbeq1a 2944	. . . . . . . . . . 11 |- ( k = x -> A = [_ x / k ]_ A )
19	18	eleq1d 2157	. . . . . . . . . 10 |- ( k = x -> ( A e. CC <-> [_ x / k ]_ A e. CC ) )
20	17, 19	rspc 2719	. . . . . . . . 9 |- ( x e. Z -> ( A. k e. Z A e. CC -> [_ x / k ]_ A e. CC ) )
21	12, 15, 20	sylc 62	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> [_ x / k ]_ A e. CC )
22	13, 21	mulcld 7571	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( B x. [_ x / k ]_ A ) e. CC )
23		nfcv 2229	. . . . . . . 8 |- F/_ k x
24		nfcv 2229	. . . . . . . . 9 |- F/_ k B
25		nfcv 2229	. . . . . . . . 9 |- F/_ k x.
26	24, 25, 16	nfov 5695	. . . . . . . 8 |- F/_ k ( B x. [_ x / k ]_ A )
27	18	oveq2d 5684	. . . . . . . 8 |- ( k = x -> ( B x. A ) = ( B x. [_ x / k ]_ A ) )
28		eqid 2089	. . . . . . . 8 |- ( k e. Z |-> ( B x. A ) ) = ( k e. Z |-> ( B x. A ) )
29	23, 26, 27, 28	fvmptf 5410	. . . . . . 7 |- ( ( x e. Z /\ ( B x. [_ x / k ]_ A ) e. CC ) -> ( ( k e. Z |-> ( B x. A ) ) ` x ) = ( B x. [_ x / k ]_ A ) )
30	12, 22, 29	syl2anc 404	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( k e. Z |-> ( B x. A ) ) ` x ) = ( B x. [_ x / k ]_ A ) )
31	30, 22	eqeltrd 2165	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( k e. Z |-> ( B x. A ) ) ` x ) e. CC )
32	2, 31	iseqseq3 9965	. . . 4 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) , CC ) = seq M ( + , ( k e. Z |-> ( B x. A ) ) ) )
33		fveq2 5320	. . . . . . . . 9 |- ( k = x -> ( F ` k ) = ( F ` x ) )
34	33	eleq1d 2157	. . . . . . . 8 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
35		isumcl.3	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
36	35, 6	eqeltrd 2165	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
37	36	ralrimiva 2447	. . . . . . . . 9 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
38	37	adantr 271	. . . . . . . 8 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. Z ( F ` k ) e. CC )
39	34, 38, 12	rspcdva 2730	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
40	2, 39	iseqseq3 9965	. . . . . 6 |- ( ph -> seq M ( + , F , CC ) = seq M ( + , F ) )
41		isumcl.5	. . . . . . 7 |- ( ph -> seq M ( + , F ) e. dom ~~> )
42	1, 2, 35, 6, 41	isumclim2 10879	. . . . . 6 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
43	40, 42	eqbrtrd 3873	. . . . 5 |- ( ph -> seq M ( + , F , CC ) ~~> sum_ k e. Z A )
44		fveq2 5320	. . . . . . 7 |- ( k = m -> ( F ` k ) = ( F ` m ) )
45	44	eleq1d 2157	. . . . . 6 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
46	37	adantr 271	. . . . . 6 |- ( ( ph /\ m e. Z ) -> A. k e. Z ( F ` k ) e. CC )
47		simpr 109	. . . . . 6 |- ( ( ph /\ m e. Z ) -> m e. Z )
48	45, 46, 47	rspcdva 2730	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( F ` m ) e. CC )
49		simpr 109	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> k e. Z )
50	28	fvmpt2 5401	. . . . . . . . 9 |- ( ( k e. Z /\ ( B x. A ) e. CC ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
51	49, 7, 50	syl2anc 404	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
52	35	oveq2d 5684	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( B x. ( F ` k ) ) = ( B x. A ) )
53	51, 52	eqtr4d 2124	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
54	53	ralrimiva 2447	. . . . . 6 |- ( ph -> A. k e. Z ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
55		nffvmpt1 5331	. . . . . . . 8 |- F/_ k ( ( k e. Z |-> ( B x. A ) ) ` m )
56	55	nfeq1 2239	. . . . . . 7 |- F/ k ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) )
57		fveq2 5320	. . . . . . . 8 |- ( k = m -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( ( k e. Z |-> ( B x. A ) ) ` m ) )
58	44	oveq2d 5684	. . . . . . . 8 |- ( k = m -> ( B x. ( F ` k ) ) = ( B x. ( F ` m ) ) )
59	57, 58	eqeq12d 2103	. . . . . . 7 |- ( k = m -> ( ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) <-> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) ) ) )
60	56, 59	rspc 2719	. . . . . 6 |- ( m e. Z -> ( A. k e. Z ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) ) ) )
61	54, 60	mpan9 276	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) ) )
62	1, 2, 4, 43, 48, 61	iisermulc2 10791	. . . 4 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) , CC ) ~~> ( B x. sum_ k e. Z A ) )
63	32, 62	eqbrtrrd 3875	. . 3 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) ) ~~> ( B x. sum_ k e. Z A ) )
64	1, 2, 3, 9, 63	isumclim 10878	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. sum_ k e. Z A ) )
65	7	ralrimiva 2447	. . 3 |- ( ph -> A. k e. Z ( B x. A ) e. CC )
66		sumfct 10826	. . 3 |- ( A. k e. Z ( B x. A ) e. CC -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = sum_ k e. Z ( B x. A ) )
67	65, 66	syl 14	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = sum_ k e. Z ( B x. A ) )
68	64, 67	eqtr3d 2123	1 |- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   A.wral 2360   [_csb 2936   |-> cmpt 3907   dom cdm 4454   ` cfv 5030  (class class class)co 5668   CCcc 7411   + caddc 7416   x. cmul 7418   ZZcz 8813   ZZ>=cuz 9082   seqcseq4 9914   seqcseq 9915   ~~> cli 10729   sum_csu 10805

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527  ax-caucvg 7528

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-frec 6172  df-1o 6197  df-oadd 6201  df-er 6308  df-en 6514  df-dom 6515  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-2 8544  df-3 8545  df-4 8546  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-rp 9198  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917  df-exp 10018  df-ihash 10247  df-cj 10339  df-re 10340  df-im 10341  df-rsqrt 10494  df-abs 10495  df-clim 10730  df-isum 10806

This theorem is referenced by:  isummulc1  10884  trirecip  10958  geoisum1c  10977