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Theorem isummulc2 13351
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 13308 . 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 2455 . . 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 465 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
7 isumcl.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
86, 7mulcld 9521 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  A )  e.  CC )
9 eqid 2454 . . . . 5  |-  ( k  e.  Z  |->  ( B  x.  A ) )  =  ( k  e.  Z  |->  ( B  x.  A ) )
108, 9fmptd 5979 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( B  x.  A
) ) : Z --> CC )
1110ffvelrnda 5955 . . 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 13347 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
1512, 7eqeltrd 2542 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1615ralrimiva 2830 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
17 fveq2 5802 . . . . . . 7  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
1817eleq1d 2523 . . . . . 6  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
1918rspccva 3178 . . . . 5  |-  ( ( A. k  e.  Z  ( F `  k )  e.  CC  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
2016, 19sylan 471 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
21 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
22 ovex 6228 . . . . . . . 8  |-  ( B  x.  A )  e. 
_V
239fvmpt2 5893 . . . . . . . 8  |-  ( ( k  e.  Z  /\  ( B  x.  A
)  e.  _V )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  A ) )
2421, 22, 23sylancl 662 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  A ) )
2512oveq2d 6219 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  A
) )
2624, 25eqtr4d 2498 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  ( F `  k ) ) )
2726ralrimiva 2830 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) ) )
28 nffvmpt1 5810 . . . . . . 7  |-  F/_ k
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )
2928nfeq1 2631 . . . . . 6  |-  F/ k ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) )
30 fveq2 5802 . . . . . . 7  |-  ( k  =  m  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
3117oveq2d 6219 . . . . . . 7  |-  ( k  =  m  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  ( F `  m )
) )
3230, 31eqeq12d 2476 . . . . . 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 3173 . . . . 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 469 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( B  x.  ( F `  m ) ) )
352, 3, 5, 14, 20, 34isermulc2 13257 . . 3  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) )  ~~>  ( B  x.  sum_ k  e.  Z  A
) )
362, 3, 4, 11, 35isumclim 13346 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  sum_ k  e.  Z  A )
)
371, 36syl5reqr 2510 1  |-  ( ph  ->  ( B  x.  sum_ k  e.  Z  A
)  =  sum_ k  e.  Z  ( B  x.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    |-> cmpt 4461   dom cdm 4951   ` cfv 5529  (class class class)co 6203   CCcc 9395    + caddc 9400    x. cmul 9402   ZZcz 10761   ZZ>=cuz 10976    seqcseq 11927    ~~> cli 13084   sum_csu 13285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-fzo 11670  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-sum 13286
This theorem is referenced by:  isummulc1  13352  trirecip  13447  geoisum1c  13462  isumneg  29946
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