Theorem isummulc2 11932

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 2230	. . 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 8163	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( B x. A ) e. CC )
8	7	fmpttd 5789	. . . 4 |- ( ph -> ( k e. Z |-> ( B x. A ) ) : Z --> CC )
9	8	ffvelcdmda 5769	. . 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 11928	. . . 4 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
13	10, 6	eqeltrd 2306	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
14	13	ralrimiva 2603	. . . . 5 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
15		fveq2 5626	. . . . . . 7 |- ( k = m -> ( F ` k ) = ( F ` m ) )
16	15	eleq1d 2298	. . . . . 6 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
17	16	rspccva 2906	. . . . 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 2229	. . . . . . . . 9 |- ( k e. Z |-> ( B x. A ) ) = ( k e. Z |-> ( B x. A ) )
21	20	fvmpt2 5717	. . . . . . . 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 6016	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( B x. ( F ` k ) ) = ( B x. A ) )
24	22, 23	eqtr4d 2265	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
25	24	ralrimiva 2603	. . . . 5 |- ( ph -> A. k e. Z ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
26		nffvmpt1 5637	. . . . . . 7 |- F/_ k ( ( k e. Z |-> ( B x. A ) ) ` m )
27	26	nfeq1 2382	. . . . . 6 |- F/ k ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) )
28		fveq2 5626	. . . . . . 7 |- ( k = m -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( ( k e. Z |-> ( B x. A ) ) ` m ) )
29	15	oveq2d 6016	. . . . . . 7 |- ( k = m -> ( B x. ( F ` k ) ) = ( B x. ( F ` m ) ) )
30	28, 29	eqeq12d 2244	. . . . . 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 2901	. . . . 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 11846	. . 3 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) ) ~~> ( B x. sum_ k e. Z A ) )
34	1, 2, 3, 9, 33	isumclim 11927	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. sum_ k e. Z A ) )
35	7	ralrimiva 2603	. . 3 |- ( ph -> A. k e. Z ( B x. A ) e. CC )
36		sumfct 11880	. . 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 2264	1 |- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   |-> cmpt 4144   dom cdm 4718   ` cfv 5317  (class class class)co 6000   CCcc 7993   + caddc 7998   x. cmul 8000   ZZcz 9442   ZZ>=cuz 9718   seqcseq 10664   ~~> cli 11784   sum_csu 11859

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-sumdc 11860

This theorem is referenced by:  isummulc1  11933  trirecip  12007  geoisum1c  12026