Theorem isermulc2 11343

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 10444	. . 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 10471	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
9	8	ffvelcdmda 5651	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F ) ` j ) e. CC )
10		addcl 7935	. . . 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 7942	. . . . 5 |- ( ( C e. CC /\ k e. CC /\ x e. CC ) -> ( C x. ( k + x ) ) = ( ( C x. k ) + ( C x. x ) ) )
14	13	3expb 1204	. . . 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 2270	. . 3 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
18	1	eleq2i 2244	. . . . 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 7937	. . . 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 10510	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G ) ` j ) = ( C x. ( seq M ( + , F ) ` j ) ) )
27	1, 2, 3, 4, 6, 9, 26	climmulc2 11334	1 |- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2737   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   + caddc 7813   x. cmul 7815   ZZcz 9251   ZZ>=cuz 9526   seqcseq 10442   ~~> cli 11281

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282

This theorem is referenced by:  isummulc2  11429  mertensabs  11540  ege2le3  11674  eftlub  11693