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Theorem isummulc2 11135
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 2116 . . 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 272 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
6 isumcl.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
75, 6mulcld 7750 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  A )  e.  CC )
87fmpttd 5541 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( B  x.  A
) ) : Z --> CC )
98ffvelrnda 5521 . . 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 11131 . . . 4  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
1310, 6eqeltrd 2192 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
1413ralrimiva 2480 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
15 fveq2 5387 . . . . . . 7  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
1615eleq1d 2184 . . . . . 6  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
1716rspccva 2760 . . . . 5  |-  ( ( A. k  e.  Z  ( F `  k )  e.  CC  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
1814, 17sylan 279 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
19 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
20 eqid 2115 . . . . . . . . 9  |-  ( k  e.  Z  |->  ( B  x.  A ) )  =  ( k  e.  Z  |->  ( B  x.  A ) )
2120fvmpt2 5470 . . . . . . . 8  |-  ( ( k  e.  Z  /\  ( B  x.  A
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `
 k )  =  ( B  x.  A
) )
2219, 7, 21syl2anc 406 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  A ) )
2310oveq2d 5756 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  A
) )
2422, 23eqtr4d 2151 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  ( F `  k ) ) )
2524ralrimiva 2480 . . . . 5  |-  ( ph  ->  A. k  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) ) )
26 nffvmpt1 5398 . . . . . . 7  |-  F/_ k
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )
2726nfeq1 2266 . . . . . 6  |-  F/ k ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) )
28 fveq2 5387 . . . . . . 7  |-  ( k  =  m  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
2915oveq2d 5756 . . . . . . 7  |-  ( k  =  m  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  ( F `  m )
) )
3028, 29eqeq12d 2130 . . . . . 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 2755 . . . . 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 277 . . . 4  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( B  x.  ( F `  m ) ) )
331, 2, 4, 12, 18, 32isermulc2 11049 . . 3  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) )  ~~>  ( B  x.  sum_ k  e.  Z  A
) )
341, 2, 3, 9, 33isumclim 11130 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  sum_ k  e.  Z  A )
)
357ralrimiva 2480 . . 3  |-  ( ph  ->  A. k  e.  Z  ( B  x.  A
)  e.  CC )
36 sumfct 11083 . . 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 2150 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 103    = wceq 1314    e. wcel 1463   A.wral 2391    |-> cmpt 3957   dom cdm 4507   ` cfv 5091  (class class class)co 5740   CCcc 7582    + caddc 7587    x. cmul 7589   ZZcz 9005   ZZ>=cuz 9275    seqcseq 10158    ~~> cli 10987   sum_csu 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  isummulc1  11136  trirecip  11210  geoisum1c  11229
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