ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isummulc2 Unicode version

Theorem isummulc2 10883
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 variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isumcl.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 isumcl.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 eqidd 2090 . . 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 271 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
6 isumcl.4 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  CC )
75, 6mulcld 7571 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  A )  e.  CC )
87fmpttd 5469 . . . 4  |-  ( ph  ->  ( k  e.  Z  |->  ( B  x.  A
) ) : Z --> CC )
98ffvelrnda 5450 . . 3  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  e.  CC )
101eleq2i 2155 . . . . . . . . 9  |-  ( x  e.  Z  <->  x  e.  ( ZZ>= `  M )
)
1110biimpri 132 . . . . . . . 8  |-  ( x  e.  ( ZZ>= `  M
)  ->  x  e.  Z )
1211adantl 272 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  Z )
134adantr 271 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  B  e.  CC )
146ralrimiva 2447 . . . . . . . . . 10  |-  ( ph  ->  A. k  e.  Z  A  e.  CC )
1514adantr 271 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  Z  A  e.  CC )
16 nfcsb1v 2966 . . . . . . . . . . 11  |-  F/_ k [_ x  /  k ]_ A
1716nfel1 2240 . . . . . . . . . 10  |-  F/ k
[_ x  /  k ]_ A  e.  CC
18 csbeq1a 2944 . . . . . . . . . . 11  |-  ( k  =  x  ->  A  =  [_ x  /  k ]_ A )
1918eleq1d 2157 . . . . . . . . . 10  |-  ( k  =  x  ->  ( A  e.  CC  <->  [_ x  / 
k ]_ A  e.  CC ) )
2017, 19rspc 2719 . . . . . . . . 9  |-  ( x  e.  Z  ->  ( A. k  e.  Z  A  e.  CC  ->  [_ x  /  k ]_ A  e.  CC )
)
2112, 15, 20sylc 62 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  [_ x  / 
k ]_ A  e.  CC )
2213, 21mulcld 7571 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( B  x.  [_ x  /  k ]_ A )  e.  CC )
23 nfcv 2229 . . . . . . . 8  |-  F/_ k
x
24 nfcv 2229 . . . . . . . . 9  |-  F/_ k B
25 nfcv 2229 . . . . . . . . 9  |-  F/_ k  x.
2624, 25, 16nfov 5695 . . . . . . . 8  |-  F/_ k
( B  x.  [_ x  /  k ]_ A
)
2718oveq2d 5684 . . . . . . . 8  |-  ( k  =  x  ->  ( B  x.  A )  =  ( B  x.  [_ x  /  k ]_ A ) )
28 eqid 2089 . . . . . . . 8  |-  ( k  e.  Z  |->  ( B  x.  A ) )  =  ( k  e.  Z  |->  ( B  x.  A ) )
2923, 26, 27, 28fvmptf 5410 . . . . . . 7  |-  ( ( x  e.  Z  /\  ( B  x.  [_ x  /  k ]_ A
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `
 x )  =  ( B  x.  [_ x  /  k ]_ A
) )
3012, 22, 29syl2anc 404 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  ( B  x.  A ) ) `  x )  =  ( B  x.  [_ x  /  k ]_ A ) )
3130, 22eqeltrd 2165 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
k  e.  Z  |->  ( B  x.  A ) ) `  x )  e.  CC )
322, 31iseqseq3 9965 . . . 4  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) ,  CC )  =  seq M (  +  , 
( k  e.  Z  |->  ( B  x.  A
) ) ) )
33 fveq2 5320 . . . . . . . . 9  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
3433eleq1d 2157 . . . . . . . 8  |-  ( k  =  x  ->  (
( F `  k
)  e.  CC  <->  ( F `  x )  e.  CC ) )
35 isumcl.3 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  A )
3635, 6eqeltrd 2165 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
3736ralrimiva 2447 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
3837adantr 271 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A. k  e.  Z  ( F `  k )  e.  CC )
3934, 38, 12rspcdva 2730 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  CC )
402, 39iseqseq3 9965 . . . . . 6  |-  ( ph  ->  seq M (  +  ,  F ,  CC )  =  seq M (  +  ,  F ) )
41 isumcl.5 . . . . . . 7  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
421, 2, 35, 6, 41isumclim2 10879 . . . . . 6  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  Z  A )
4340, 42eqbrtrd 3873 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  sum_ k  e.  Z  A )
44 fveq2 5320 . . . . . . 7  |-  ( k  =  m  ->  ( F `  k )  =  ( F `  m ) )
4544eleq1d 2157 . . . . . 6  |-  ( k  =  m  ->  (
( F `  k
)  e.  CC  <->  ( F `  m )  e.  CC ) )
4637adantr 271 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  A. k  e.  Z  ( F `  k )  e.  CC )
47 simpr 109 . . . . . 6  |-  ( (
ph  /\  m  e.  Z )  ->  m  e.  Z )
4845, 46, 47rspcdva 2730 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  ( F `  m )  e.  CC )
49 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  k  e.  Z )
5028fvmpt2 5401 . . . . . . . . 9  |-  ( ( k  e.  Z  /\  ( B  x.  A
)  e.  CC )  ->  ( ( k  e.  Z  |->  ( B  x.  A ) ) `
 k )  =  ( B  x.  A
) )
5149, 7, 50syl2anc 404 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  A ) )
5235oveq2d 5684 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  A
) )
5351, 52eqtr4d 2124 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( B  x.  ( F `  k ) ) )
5453ralrimiva 2447 . . . . . 6  |-  ( ph  ->  A. k  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) ) )
55 nffvmpt1 5331 . . . . . . . 8  |-  F/_ k
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )
5655nfeq1 2239 . . . . . . 7  |-  F/ k ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) )
57 fveq2 5320 . . . . . . . 8  |-  ( k  =  m  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  k
)  =  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m ) )
5844oveq2d 5684 . . . . . . . 8  |-  ( k  =  m  ->  ( B  x.  ( F `  k ) )  =  ( B  x.  ( F `  m )
) )
5957, 58eqeq12d 2103 . . . . . . 7  |-  ( k  =  m  ->  (
( ( k  e.  Z  |->  ( B  x.  A ) ) `  k )  =  ( B  x.  ( F `
 k ) )  <-> 
( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  ( F `
 m ) ) ) )
6056, 59rspc 2719 . . . . . 6  |-  ( 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 )
) ) )
6154, 60mpan9 276 . . . . 5  |-  ( (
ph  /\  m  e.  Z )  ->  (
( k  e.  Z  |->  ( B  x.  A
) ) `  m
)  =  ( B  x.  ( F `  m ) ) )
621, 2, 4, 43, 48, 61iisermulc2 10791 . . . 4  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) ,  CC )  ~~>  ( B  x.  sum_ k  e.  Z  A ) )
6332, 62eqbrtrrd 3875 . . 3  |-  ( ph  ->  seq M (  +  ,  ( k  e.  Z  |->  ( B  x.  A ) ) )  ~~>  ( B  x.  sum_ k  e.  Z  A
) )
641, 2, 3, 9, 63isumclim 10878 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  ( B  x.  sum_ k  e.  Z  A )
)
657ralrimiva 2447 . . 3  |-  ( ph  ->  A. k  e.  Z  ( B  x.  A
)  e.  CC )
66 sumfct 10826 . . 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
) )
6765, 66syl 14 . 2  |-  ( ph  -> 
sum_ m  e.  Z  ( ( k  e.  Z  |->  ( B  x.  A ) ) `  m )  =  sum_ k  e.  Z  ( B  x.  A )
)
6864, 67eqtr3d 2123 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 1290    e. wcel 1439   A.wral 2360   [_csb 2936    |-> cmpt 3907   dom cdm 4454   ` cfv 5030  (class class class)co 5668   CCcc 7411    + caddc 7416    x. cmul 7418   ZZcz 8813   ZZ>=cuz 9082    seqcseq4 9914    seqcseq 9915    ~~> cli 10729   sum_csu 10805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527  ax-caucvg 7528
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-frec 6172  df-1o 6197  df-oadd 6201  df-er 6308  df-en 6514  df-dom 6515  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-2 8544  df-3 8545  df-4 8546  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-rp 9198  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917  df-exp 10018  df-ihash 10247  df-cj 10339  df-re 10340  df-im 10341  df-rsqrt 10494  df-abs 10495  df-clim 10730  df-isum 10806
This theorem is referenced by:  isummulc1  10884  trirecip  10958  geoisum1c  10977
  Copyright terms: Public domain W3C validator