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Theorem iisermulc2 10305
Description: Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
isermulc2.2  |-  ( ph  ->  M  e.  ZZ )
isermulc2.4  |-  ( ph  ->  C  e.  CC )
isermulc2.5  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  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
iisermulc2  |-  ( ph  ->  seq M (  +  ,  G ,  CC ) 
~~>  ( C  x.  A
) )
Distinct variable groups:    A, k    k, F    k, M    C, k    k, G    ph, k    k, Z

Proof of Theorem iisermulc2
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clim2ser.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 isermulc2.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 isermulc2.5 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
4 isermulc2.4 . 2  |-  ( ph  ->  C  e.  CC )
5 iseqex 9512 . . 3  |-  seq M
(  +  ,  G ,  CC )  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  G ,  CC )  e.  _V )
7 isermulc2.6 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
81, 2, 7iserf 9538 . . 3  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : Z --> CC )
98ffvelrnda 5328 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ,  CC ) `  j )  e.  CC )
10 addcl 7149 . . . 4  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
1110adantl 271 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  +  x )  e.  CC )
124adantr 270 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  C  e.  CC )
13 adddi 7156 . . . . 5  |-  ( ( C  e.  CC  /\  k  e.  CC  /\  x  e.  CC )  ->  ( C  x.  ( k  +  x ) )  =  ( ( C  x.  k )  +  ( C  x.  x ) ) )
14133expb 1140 . . . 4  |-  ( ( C  e.  CC  /\  ( k  e.  CC  /\  x  e.  CC ) )  ->  ( C  x.  ( k  +  x
) )  =  ( ( C  x.  k
)  +  ( C  x.  x ) ) )
1512, 14sylan 277 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( C  x.  ( k  +  x
) )  =  ( ( C  x.  k
)  +  ( C  x.  x ) ) )
16 simpr 108 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
1716, 1syl6eleq 2172 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
181eleq2i 2146 . . . . 5  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
1918, 7sylan2br 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
2019adantlr 461 . . 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 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( C  x.  ( F `
 k ) ) )
2322adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( C  x.  ( F `
 k ) ) )
24 cnex 7148 . . . 4  |-  CC  e.  _V
2524a1i 9 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  CC  e.  _V )
26 mulcl 7151 . . . 4  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
2726adantl 271 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  x.  x )  e.  CC )
284adantr 270 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  C  e.  CC )
2928, 7mulcld 7190 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( C  x.  ( F `  k ) )  e.  CC )
3021, 29eqeltrd 2156 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
3118, 30sylan2br 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3231adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3311, 15, 17, 20, 23, 25, 27, 32, 12iseqdistr 9556 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ,  CC ) `  j )  =  ( C  x.  (  seq M (  +  ,  F ,  CC ) `  j ) ) )
341, 2, 3, 4, 6, 9, 33climmulc2 10296 1  |-  ( ph  ->  seq M (  +  ,  G ,  CC ) 
~~>  ( C  x.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   CCcc 7030    + caddc 7035    x. cmul 7037   ZZcz 8421   ZZ>=cuz 8689    seqcseq 9510    ~~> cli 10244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145  ax-arch 7146  ax-caucvg 7147
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-if 3354  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-frec 6034  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-inn 8096  df-2 8154  df-3 8155  df-4 8156  df-n0 8345  df-z 8422  df-uz 8690  df-rp 8805  df-iseq 9511  df-iexp 9562  df-cj 9856  df-re 9857  df-im 9858  df-rsqrt 10011  df-abs 10012  df-clim 10245
This theorem is referenced by: (None)
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