Theorem isummulc2 11466

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 variable m is 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 2190	. . 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 276	. . . . . 6 |- ( ( ph /\ k e. Z ) -> B e. CC )
6		isumcl.4	. . . . . 6 |- ( ( ph /\ k e. Z ) -> A e. CC )
7	5, 6	mulcld 8008	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( B x. A ) e. CC )
8	7	fmpttd 5692	. . . 4 |- ( ph -> ( k e. Z |-> ( B x. A ) ) : Z --> CC )
9	8	ffvelcdmda 5672	. . 3 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) e. CC )
10		isumcl.3	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
11		isumcl.5	. . . . 5 |- ( ph -> seq M ( + , F ) e. dom ~~> )
12	1, 2, 10, 6, 11	isumclim2 11462	. . . 4 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
13	10, 6	eqeltrd 2266	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
14	13	ralrimiva 2563	. . . . 5 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
15		fveq2 5534	. . . . . . 7 |- ( k = m -> ( F ` k ) = ( F ` m ) )
16	15	eleq1d 2258	. . . . . 6 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
17	16	rspccva 2855	. . . . 5 |- ( ( A. k e. Z ( F ` k ) e. CC /\ m e. Z ) -> ( F ` m ) e. CC )
18	14, 17	sylan 283	. . . 4 |- ( ( ph /\ m e. Z ) -> ( F ` m ) e. CC )
19		simpr 110	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> k e. Z )
20		eqid 2189	. . . . . . . . 9 |- ( k e. Z |-> ( B x. A ) ) = ( k e. Z |-> ( B x. A ) )
21	20	fvmpt2 5620	. . . . . . . 8 |- ( ( k e. Z /\ ( B x. A ) e. CC ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
22	19, 7, 21	syl2anc 411	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
23	10	oveq2d 5912	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( B x. ( F ` k ) ) = ( B x. A ) )
24	22, 23	eqtr4d 2225	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
25	24	ralrimiva 2563	. . . . 5 |- ( ph -> A. k e. Z ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
26		nffvmpt1 5545	. . . . . . 7 |- F/_ k ( ( k e. Z |-> ( B x. A ) ) ` m )
27	26	nfeq1 2342	. . . . . 6 |- F/ k ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) )
28		fveq2 5534	. . . . . . 7 |- ( k = m -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( ( k e. Z |-> ( B x. A ) ) ` m ) )
29	15	oveq2d 5912	. . . . . . 7 |- ( k = m -> ( B x. ( F ` k ) ) = ( B x. ( F ` m ) ) )
30	28, 29	eqeq12d 2204	. . . . . 6 |- ( k = m -> ( ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) <-> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) ) ) )
31	27, 30	rspc 2850	. . . . 5 |- ( 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 ) ) ) )
32	25, 31	mpan9 281	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) ) )
33	1, 2, 4, 12, 18, 32	isermulc2 11380	. . 3 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) ) ~~> ( B x. sum_ k e. Z A ) )
34	1, 2, 3, 9, 33	isumclim 11461	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. sum_ k e. Z A ) )
35	7	ralrimiva 2563	. . 3 |- ( ph -> A. k e. Z ( B x. A ) e. CC )
36		sumfct 11414	. . 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 ) )
37	35, 36	syl 14	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = sum_ k e. Z ( B x. A ) )
38	34, 37	eqtr3d 2224	1 |- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   |-> cmpt 4079   dom cdm 4644   ` cfv 5235  (class class class)co 5896   CCcc 7839   + caddc 7844   x. cmul 7846   ZZcz 9283   ZZ>=cuz 9558   seqcseq 10476   ~~> cli 11318   sum_csu 11393

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-frec 6416  df-1o 6441  df-oadd 6445  df-er 6559  df-en 6767  df-dom 6768  df-fin 6769  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-seqfrec 10477  df-exp 10551  df-ihash 10788  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319  df-sumdc 11394

This theorem is referenced by:  isummulc1  11467  trirecip  11541  geoisum1c  11560