Theorem isermulc2 11281

Description: Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)

Hypotheses

Ref	Expression
clim2iser.1	|- Z = ( ZZ>= ` M )
isermulc2.2	|- ( ph -> M e. ZZ )
isermulc2.4	|- ( ph -> C e. CC )
isermulc2.5	|- ( ph -> seq M ( + , F ) ~~> A )
isermulc2.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
isermulc2.7	|- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C x. ( F ` k ) ) )

Assertion

Ref	Expression
isermulc2	|- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )

Distinct variable groups:   A, k   k, F   k, M   C, k   k, G   ph, k   k, Z

Proof of Theorem isermulc2

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

Step	Hyp	Ref	Expression
1		clim2iser.1	. 2 |- Z = ( ZZ>= ` M )
2		isermulc2.2	. 2 |- ( ph -> M e. ZZ )
3		isermulc2.5	. 2 |- ( ph -> seq M ( + , F ) ~~> A )
4		isermulc2.4	. 2 |- ( ph -> C e. CC )
5		seqex 10382	. . 3 |- seq M ( + , G ) e. _V
6	5	a1i 9	. 2 |- ( ph -> seq M ( + , G ) e. _V )
7		isermulc2.6	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
8	1, 2, 7	serf 10409	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
9	8	ffvelrnda 5620	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
10		addcl 7878	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
11	10	adantl 275	. . 3 |- ( ( ( ph /\ j e. Z ) /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
12	4	adantr 274	. . . 4 |- ( ( ph /\ j e. Z ) -> C e. CC )
13		adddi 7885	. . . . 5 |- ( ( C e. CC /\ k e. CC /\ x e. CC ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
14	13	3expb 1194	. . . 4 |- ( ( C e. CC /\ ( k e. CC /\ x e. CC ) ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
15	12, 14	sylan 281	. . 3 |- ( ( ( ph /\ j e. Z ) /\ ( k e. CC /\ x e. CC ) ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
16		simpr 109	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. Z )
17	16, 1	eleqtrdi 2259	. . 3 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
18	1	eleq2i 2233	. . . . 5 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
19	18, 7	sylan2br 286	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
20	19	adantlr 469	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
21		isermulc2.7	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C x. ( F ` k ) ) )
22	18, 21	sylan2br 286	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( C x. ( F ` k ) ) )
23	22	adantlr 469	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( C x. ( F ` k ) ) )
24		mulcl 7880	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
25	24	adantl 275	. . 3 |- ( ( ( ph /\ j e. Z ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
26	11, 15, 17, 20, 23, 25, 12	seq3distr 10448	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) = ( C x. ( seq M ( + , F ) ` j ) ) )
27	1, 2, 3, 4, 6, 9, 26	climmulc2 11272	1 |- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   + caddc 7756   x. cmul 7758   ZZcz 9191   ZZ>=cuz 9466   seqcseq 10380   ~~> cli 11219

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220

This theorem is referenced by:  isummulc2  11367  mertensabs  11478  ege2le3  11612  eftlub  11631