Theorem isummulc2 11195

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 2140	. . 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 274	. . . . . 6 |- ( ( ph /\ k e. Z ) -> B e. CC )
6		isumcl.4	. . . . . 6 |- ( ( ph /\ k e. Z ) -> A e. CC )
7	5, 6	mulcld 7786	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( B x. A ) e. CC )
8	7	fmpttd 5575	. . . 4 |- ( ph -> ( k e. Z |-> ( B x. A ) ) : Z --> CC )
9	8	ffvelrnda 5555	. . 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 11191	. . . 4 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
13	10, 6	eqeltrd 2216	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
14	13	ralrimiva 2505	. . . . 5 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
15		fveq2 5421	. . . . . . 7 |- ( k = m -> ( F ` k ) = ( F ` m ) )
16	15	eleq1d 2208	. . . . . 6 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
17	16	rspccva 2788	. . . . 5 |- ( ( A. k e. Z ( F ` k ) e. CC /\ m e. Z ) -> ( F ` m ) e. CC )
18	14, 17	sylan 281	. . . 4 |- ( ( ph /\ m e. Z ) -> ( F ` m ) e. CC )
19		simpr 109	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> k e. Z )
20		eqid 2139	. . . . . . . . 9 |- ( k e. Z |-> ( B x. A ) ) = ( k e. Z |-> ( B x. A ) )
21	20	fvmpt2 5504	. . . . . . . 8 |- ( ( k e. Z /\ ( B x. A ) e. CC ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
22	19, 7, 21	syl2anc 408	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
23	10	oveq2d 5790	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( B x. ( F ` k ) ) = ( B x. A ) )
24	22, 23	eqtr4d 2175	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
25	24	ralrimiva 2505	. . . . 5 |- ( ph -> A. k e. Z ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
26		nffvmpt1 5432	. . . . . . 7 |- F/_ k ( ( k e. Z |-> ( B x. A ) ) ` m )
27	26	nfeq1 2291	. . . . . 6 |- F/ k ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) )
28		fveq2 5421	. . . . . . 7 |- ( k = m -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( ( k e. Z |-> ( B x. A ) ) ` m ) )
29	15	oveq2d 5790	. . . . . . 7 |- ( k = m -> ( B x. ( F ` k ) ) = ( B x. ( F ` m ) ) )
30	28, 29	eqeq12d 2154	. . . . . 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 2783	. . . . 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 279	. . . 4 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) ) )
33	1, 2, 4, 12, 18, 32	isermulc2 11109	. . 3 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) ) ~~> ( B x. sum_ k e. Z A ) )
34	1, 2, 3, 9, 33	isumclim 11190	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. sum_ k e. Z A ) )
35	7	ralrimiva 2505	. . 3 |- ( ph -> A. k e. Z ( B x. A ) e. CC )
36		sumfct 11143	. . 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 2174	1 |- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2416   |-> cmpt 3989   dom cdm 4539   ` cfv 5123  (class class class)co 5774   CCcc 7618   + caddc 7623   x. cmul 7625   ZZcz 9054   ZZ>=cuz 9326   seqcseq 10218   ~~> cli 11047   sum_csu 11122

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  isummulc1  11196  trirecip  11270  geoisum1c  11289