Theorem isummulc2 12171

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 2239	. . 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 8336	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( B x. A ) e. CC )
8	7	fmpttd 5854	. . . 4 |- ( ph -> ( k e. Z |-> ( B x. A ) ) : Z --> CC )
9	8	ffvelcdmda 5834	. . 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 12167	. . . 4 |- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A )
13	10, 6	eqeltrd 2315	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
14	13	ralrimiva 2623	. . . . 5 |- ( ph -> A. k e. Z ( F ` k ) e. CC )
15		fveq2 5690	. . . . . . 7 |- ( k = m -> ( F ` k ) = ( F ` m ) )
16	15	eleq1d 2307	. . . . . 6 |- ( k = m -> ( ( F ` k ) e. CC <-> ( F ` m ) e. CC ) )
17	16	rspccva 2928	. . . . 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 2238	. . . . . . . . 9 |- ( k e. Z |-> ( B x. A ) ) = ( k e. Z |-> ( B x. A ) )
21	20	fvmpt2 5783	. . . . . . . 8 |- ( ( k e. Z /\ ( B x. A ) e. CC ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
22	19, 7, 21	syl2anc 415	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. A ) )
23	10	oveq2d 6091	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( B x. ( F ` k ) ) = ( B x. A ) )
24	22, 23	eqtr4d 2274	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
25	24	ralrimiva 2623	. . . . 5 |- ( ph -> A. k e. Z ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( B x. ( F ` k ) ) )
26		nffvmpt1 5701	. . . . . . 7 |- F/_ k ( ( k e. Z |-> ( B x. A ) ) ` m )
27	26	nfeq1 2402	. . . . . 6 |- F/ k ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. ( F ` m ) )
28		fveq2 5690	. . . . . . 7 |- ( k = m -> ( ( k e. Z |-> ( B x. A ) ) ` k ) = ( ( k e. Z |-> ( B x. A ) ) ` m ) )
29	15	oveq2d 6091	. . . . . . 7 |- ( k = m -> ( B x. ( F ` k ) ) = ( B x. ( F ` m ) ) )
30	28, 29	eqeq12d 2253	. . . . . 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 2923	. . . . 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 12084	. . 3 |- ( ph -> seq M ( + , ( k e. Z |-> ( B x. A ) ) ) ~~> ( B x. sum_ k e. Z A ) )
34	1, 2, 3, 9, 33	isumclim 12166	. 2 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> ( B x. A ) ) ` m ) = ( B x. sum_ k e. Z A ) )
35	7	ralrimiva 2623	. . 3 |- ( ph -> A. k e. Z ( B x. A ) e. CC )
36		sumfct 12118	. . 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 2273	1 |- ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   |-> cmpt 4187   dom cdm 4769   ` cfv 5372  (class class class)co 6075   CCcc 8167   + caddc 8172   x. cmul 8174   ZZcz 9623   ZZ>=cuz 9900   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:  isummulc1  12172  trirecip  12246  geoisum1c  12265