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Theorem isummulc2 11466
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 isumcl.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 isumcl.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 eqidd 2190 . . 3  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
4 summulc.6 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
54adantr 276 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
6 isumcl.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
75, 6mulcld 8008 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  A )  e.  CC )
87fmpttd 5692 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( B  x.  A
) ) : Z --> CC )
98ffvelcdmda 5672 . . 3  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  e.  CC )
10 isumcl.3 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
11 isumcl.5 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
121, 2, 10, 6, 11isumclim2 11462 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
1310, 6eqeltrd 2266 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1413ralrimiva 2563 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
15 fveq2 5534 . . . . . . 7  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
1615eleq1d 2258 . . . . . 6  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
1716rspccva 2855 . . . . 5  |-  ( ( A. k  e.  Z  ( F `  k )  e.  CC  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
1814, 17sylan 283 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
19 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
20 eqid 2189 . . . . . . . . 9  |-  ( k  e.  Z  |->  ( B  x.  A ) )  =  ( k  e.  Z  |->  ( B  x.  A ) )
2120fvmpt2 5620 . . . . . . . 8  |-  ( ( k  e.  Z  /\  ( B  x.  A
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `
 k )  =  ( B  x.  A
) )
2219, 7, 21syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  A ) )
2310oveq2d 5912 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  A
) )
2422, 23eqtr4d 2225 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  ( F `  k ) ) )
2524ralrimiva 2563 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) ) )
26 nffvmpt1 5545 . . . . . . 7  |-  F/_ k
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )
2726nfeq1 2342 . . . . . 6  |-  F/ k ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) )
28 fveq2 5534 . . . . . . 7  |-  ( k  =  m  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
2915oveq2d 5912 . . . . . . 7  |-  ( k  =  m  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  ( F `  m )
) )
3028, 29eqeq12d 2204 . . . . . 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 ) ) ) )
3127, 30rspc 2850 . . . . 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 )
) ) )
3225, 31mpan9 281 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( B  x.  ( F `  m ) ) )
331, 2, 4, 12, 18, 32isermulc2 11380 . . 3  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) )  ~~>  ( B  x.  sum_ k  e.  Z  A
) )
341, 2, 3, 9, 33isumclim 11461 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  sum_ k  e.  Z  A )
)
357ralrimiva 2563 . . 3  |-  ( ph  ->  A. k  e.  Z  ( B  x.  A
)  e.  CC )
36 sumfct 11414 . . 3  |-  ( A. k  e.  Z  ( B  x.  A )  e.  CC  ->  sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `
 m )  = 
sum_ k  e.  Z  ( B  x.  A
) )
3735, 36syl 14 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  sum_ k  e.  Z  ( B  x.  A )
)
3834, 37eqtr3d 2224 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 104    = wceq 1364    e. wcel 2160   A.wral 2468    |-> cmpt 4079   dom cdm 4644   ` cfv 5235  (class class class)co 5896   CCcc 7839    + caddc 7844    x. cmul 7846   ZZcz 9283   ZZ>=cuz 9558    seqcseq 10476    ~~> cli 11318   sum_csu 11393
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-frec 6416  df-1o 6441  df-oadd 6445  df-er 6559  df-en 6767  df-dom 6768  df-fin 6769  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-seqfrec 10477  df-exp 10551  df-ihash 10788  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319  df-sumdc 11394
This theorem is referenced by:  isummulc1  11467  trirecip  11541  geoisum1c  11560
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