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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.
StepHypRef 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
65a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  G )  e. 
_V )
7 isermulc2.6 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
81, 2, 7serf 10471 . . 3  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
98ffvelcdmda 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 )
1110adantl 277 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  +  x )  e.  CC )
124adantr 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 ) ) )
14133expb 1204 . . . 4  |-  ( ( C  e.  CC  /\  ( k  e.  CC  /\  x  e.  CC ) )  ->  ( C  x.  ( k  +  x
) )  =  ( ( C  x.  k
)  +  ( C  x.  x ) ) )
1512, 14sylan 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 )
1716, 1eleqtrdi 2270 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
181eleq2i 2244 . . . . 5  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
1918, 7sylan2br 288 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
2019adantlr 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 ) ) )
2218, 21sylan2br 288 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( C  x.  ( F `
 k ) ) )
2322adantlr 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 )
2524adantl 277 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  x.  x )  e.  CC )
2611, 15, 17, 20, 23, 25, 12seq3distr 10510 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ) `  j
)  =  ( C  x.  (  seq M
(  +  ,  F
) `  j )
) )
271, 2, 3, 4, 6, 9, 26climmulc2 11334 1  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  ( C  x.  A ) )
Colors of variables: wff set class
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
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