Theorem isermulc2 12131

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 10907	. . 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 10941	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
9	8	ffvelcdmda 5846	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
10		addcl 8310	. . . 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 8317	. . . . 5 |- ( ( C e. CC /\ k e. CC /\ x e. CC ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
14	13	3expb 1235	. . . 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 2331	. . 3 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
18	1	eleq2i 2305	. . . . 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 481	. . 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 481	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( C x. ( F ` k ) ) )
24		mulcl 8312	. . . 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 10990	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) = ( C x. ( seq M ( + , F ) ` j ) ) )
27	1, 2, 3, 4, 6, 9, 26	climmulc2 12122	1 |- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   + caddc 8188   x. cmul 8190   ZZcz 9655   ZZ>=cuz 9937   seqcseq 10905   ~~> cli 12069

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070

This theorem is used by:  isummulc2  12218  mertensabs  12329  ege2le3  12463  eftlub  12482  log2tlbndlog2  16220