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Theorem isummulc2 12473
Description: An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
Hypotheses
Ref Expression
isumcl.1  |-  Z  =  ( ZZ>= `  M )
isumcl.2  |-  ( ph  ->  M  e.  ZZ )
isumcl.3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
isumcl.4  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
isumcl.5  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
summulc.6  |-  ( ph  ->  B  e.  CC )
Assertion
Ref Expression
isummulc2  |-  ( ph  ->  ( B  x.  sum_ k  e.  Z  A
)  =  sum_ k  e.  Z  ( B  x.  A ) )
Distinct variable groups:    B, k    k, F    ph, k    k, Z   
k, M
Allowed substitution hint:    A( k)

Proof of Theorem isummulc2
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 sumfc 12430 . 2  |-  sum_ m  e.  Z  ( (
k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  sum_ k  e.  Z  ( B  x.  A
)
2 isumcl.1 . . 3  |-  Z  =  ( ZZ>= `  M )
3 isumcl.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
4 eqidd 2388 . . 3  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
5 summulc.6 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
65adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
7 isumcl.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
86, 7mulcld 9041 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  A )  e.  CC )
9 eqid 2387 . . . . 5  |-  ( k  e.  Z  |->  ( B  x.  A ) )  =  ( k  e.  Z  |->  ( B  x.  A ) )
108, 9fmptd 5832 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( B  x.  A
) ) : Z --> CC )
1110ffvelrnda 5809 . . 3  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  e.  CC )
12 isumcl.3 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
13 isumcl.5 . . . . 5  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
142, 3, 12, 7, 13isumclim2 12469 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
1512, 7eqeltrd 2461 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1615ralrimiva 2732 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
17 fveq2 5668 . . . . . . 7  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
1817eleq1d 2453 . . . . . 6  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
1918rspccva 2994 . . . . 5  |-  ( ( A. k  e.  Z  ( F `  k )  e.  CC  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
2016, 19sylan 458 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
21 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
22 ovex 6045 . . . . . . . 8  |-  ( B  x.  A )  e. 
_V
239fvmpt2 5751 . . . . . . . 8  |-  ( ( k  e.  Z  /\  ( B  x.  A
)  e.  _V )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  A ) )
2421, 22, 23sylancl 644 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  A ) )
2512oveq2d 6036 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  A
) )
2624, 25eqtr4d 2422 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  ( F `  k ) ) )
2726ralrimiva 2732 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) ) )
28 nffvmpt1 5676 . . . . . . 7  |-  F/_ k
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )
2928nfeq1 2532 . . . . . 6  |-  F/ k ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) )
30 fveq2 5668 . . . . . . 7  |-  ( k  =  m  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
3117oveq2d 6036 . . . . . . 7  |-  ( k  =  m  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  ( F `  m )
) )
3230, 31eqeq12d 2401 . . . . . 6  |-  ( k  =  m  ->  (
( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) )  <-> 
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) ) ) )
3329, 32rspc 2989 . . . . 5  |-  ( m  e.  Z  ->  ( A. k  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `
 m )  =  ( B  x.  ( F `  m )
) ) )
3427, 33mpan9 456 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( B  x.  ( F `  m ) ) )
352, 3, 5, 14, 20, 34isermulc2 12378 . . 3  |-  ( ph  ->  seq  M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) )  ~~>  ( B  x.  sum_ k  e.  Z  A
) )
362, 3, 4, 11, 35isumclim 12468 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  sum_ k  e.  Z  A )
)
371, 36syl5reqr 2434 1  |-  ( ph  ->  ( B  x.  sum_ k  e.  Z  A
)  =  sum_ k  e.  Z  ( B  x.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    e. cmpt 4207   dom cdm 4818   ` cfv 5394  (class class class)co 6020   CCcc 8921    + caddc 8926    x. cmul 8928   ZZcz 10214   ZZ>=cuz 10420    seq cseq 11250    ~~> cli 12205   sum_csu 12406
This theorem is referenced by:  isummulc1  12474  trirecip  12569  geoisum1c  12584  isumneg  27396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407
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