Theorem isermulc2 11846

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 10666	. . 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 10700	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
9	8	ffvelcdmda 5769	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
10		addcl 8120	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
11	10	adantl 277	. . 3 |- ( ( ( ph /\ j e. Z ) /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
12	4	adantr 276	. . . 4 |- ( ( ph /\ j e. Z ) -> C e. CC )
13		adddi 8127	. . . . 5 |- ( ( C e. CC /\ k e. CC /\ x e. CC ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
14	13	3expb 1228	. . . 4 |- ( ( C e. CC /\ ( k e. CC /\ x e. CC ) ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
15	12, 14	sylan 283	. . 3 |- ( ( ( ph /\ j e. Z ) /\ ( k e. CC /\ x e. CC ) ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
16		simpr 110	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. Z )
17	16, 1	eleqtrdi 2322	. . 3 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
18	1	eleq2i 2296	. . . . 5 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
19	18, 7	sylan2br 288	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
20	19	adantlr 477	. . 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 288	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( C x. ( F ` k ) ) )
23	22	adantlr 477	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( C x. ( F ` k ) ) )
24		mulcl 8122	. . . 4 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
25	24	adantl 277	. . 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 10749	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) = ( C x. ( seq M ( + , F ) ` j ) ) )
27	1, 2, 3, 4, 6, 9, 26	climmulc2 11837	1 |- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   + caddc 7998   x. cmul 8000   ZZcz 9442   ZZ>=cuz 9718   seqcseq 10664   ~~> cli 11784

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-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  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-rp 9846  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785

This theorem is referenced by:  isummulc2  11932  mertensabs  12043  ege2le3  12177  eftlub  12196