MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumfsum Structured version   Unicode version

Theorem gsumfsum 16756
Description: Relate a group sum on ℂfld to a finite sum on the complexes. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
gsumfsum.1  |-  ( ph  ->  A  e.  Fin )
gsumfsum.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
Assertion
Ref Expression
gsumfsum  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem gsumfsum
Dummy variables  f  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mpteq1 4281 . . . . . . 7  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  ( k  e.  (/)  |->  B ) )
2 mpt0 5564 . . . . . . 7  |-  ( k  e.  (/)  |->  B )  =  (/)
31, 2syl6eq 2483 . . . . . 6  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  (/) )
43oveq2d 6089 . . . . 5  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (fld 
gsumg  (/) ) )
5 cnfld0 16715 . . . . . . 7  |-  0  =  ( 0g ` fld )
65gsum0 14770 . . . . . 6  |-  (fld  gsumg  (/) )  =  0
7 sum0 12505 . . . . . 6  |-  sum_ k  e.  (/)  B  =  0
86, 7eqtr4i 2458 . . . . 5  |-  (fld  gsumg  (/) )  =  sum_ k  e.  (/)  B
94, 8syl6eq 2483 . . . 4  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  (/)  B )
10 sumeq1 12473 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
119, 10eqtr4d 2470 . . 3  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
1211a1i 11 . 2  |-  ( ph  ->  ( A  =  (/)  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
13 cnfldbas 16697 . . . . . . 7  |-  CC  =  ( Base ` fld )
14 cnfldadd 16698 . . . . . . 7  |-  +  =  ( +g  ` fld )
15 eqid 2435 . . . . . . 7  |-  (Cntz ` fld )  =  (Cntz ` fld )
16 cnrng 16713 . . . . . . . 8  |-fld  e.  Ring
17 rngmnd 15663 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1816, 17mp1i 12 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->fld  e.  Mnd )
19 gsumfsum.1 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
2019adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  A  e.  Fin )
21 gsumfsum.2 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
22 eqid 2435 . . . . . . . . 9  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
2321, 22fmptd 5885 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
2423adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
k  e.  A  |->  B ) : A --> CC )
25 rngcmn 15684 . . . . . . . . 9  |-  (fld  e.  Ring  ->fld  e. CMnd )
2616, 25mp1i 12 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->fld  e. CMnd )
2713, 15, 26, 24cntzcmnf 15505 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ran  ( k  e.  A  |->  B )  C_  (
(Cntz ` fld ) `  ran  (
k  e.  A  |->  B ) ) )
28 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( # `
 A )  e.  NN )
29 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
30 f1of1 5665 . . . . . . . 8  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-1-1-> A )
3129, 30syl 16 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-> A )
32 cnvimass 5216 . . . . . . . . 9  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { 0 } ) )  C_  dom  ( k  e.  A  |->  B )
33 fdm 5587 . . . . . . . . . 10  |-  ( ( k  e.  A  |->  B ) : A --> CC  ->  dom  ( k  e.  A  |->  B )  =  A )
3424, 33syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  dom  ( k  e.  A  |->  B )  =  A )
3532, 34syl5sseq 3388 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( `' ( k  e.  A  |->  B ) "
( _V  \  {
0 } ) ) 
C_  A )
36 f1ofo 5673 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-onto-> A )
37 forn 5648 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -onto-> A  ->  ran  f  =  A
)
3829, 36, 373syl 19 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ran  f  =  A )
3935, 38sseqtr4d 3377 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( `' ( k  e.  A  |->  B ) "
( _V  \  {
0 } ) ) 
C_  ran  f )
40 eqid 2435 . . . . . . 7  |-  ( `' ( ( k  e.  A  |->  B )  o.  f ) " ( _V  \  { 0 } ) )  =  ( `' ( ( k  e.  A  |->  B )  o.  f ) "
( _V  \  {
0 } ) )
4113, 5, 14, 15, 18, 20, 24, 27, 28, 31, 39, 40gsumval3 15504 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (  seq  1 (  +  ,  ( ( k  e.  A  |->  B )  o.  f ) ) `  ( # `  A ) ) )
42 sumfc 12493 . . . . . . 7  |-  sum_ x  e.  A  ( (
k  e.  A  |->  B ) `  x )  =  sum_ k  e.  A  B
43 fveq2 5720 . . . . . . . 8  |-  ( x  =  ( f `  n )  ->  (
( k  e.  A  |->  B ) `  x
)  =  ( ( k  e.  A  |->  B ) `  ( f `
 n ) ) )
4424ffvelrnda 5862 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  A )  ->  ( ( k  e.  A  |->  B ) `  x )  e.  CC )
45 f1of 5666 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) ) --> A )
4629, 45syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) --> A )
47 fvco3 5792 . . . . . . . . 9  |-  ( ( f : ( 1 ... ( # `  A
) ) --> A  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  n )  =  ( ( k  e.  A  |->  B ) `  (
f `  n )
) )
4846, 47sylan 458 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  n )  =  ( ( k  e.  A  |->  B ) `  (
f `  n )
) )
4943, 28, 29, 44, 48fsum 12504 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  sum_ x  e.  A  ( (
k  e.  A  |->  B ) `  x )  =  (  seq  1
(  +  ,  ( ( k  e.  A  |->  B )  o.  f
) ) `  ( # `
 A ) ) )
5042, 49syl5eqr 2481 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  B  =  (  seq  1 (  +  ,  ( ( k  e.  A  |->  B )  o.  f ) ) `
 ( # `  A
) ) )
5141, 50eqtr4d 2470 . . . . 5  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
5251expr 599 . . . 4  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
5352exlimdv 1646 . . 3  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( E. f 
f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
5453expimpd 587 . 2  |-  ( ph  ->  ( ( ( # `  A )  e.  NN  /\ 
E. f  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
55 fz1f1o 12494 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
5619, 55syl 16 . 2  |-  ( ph  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
5712, 54, 56mpjaod 371 1  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309   (/)c0 3620   {csn 3806    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    o. ccom 4874   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8978   0cc0 8980   1c1 8981    + caddc 8983   NNcn 9990   ...cfz 11033    seq cseq 11313   #chash 11608   sum_csu 12469    gsumg cgsu 13714   Mndcmnd 14674  Cntzccntz 15104  CMndccmn 15402   Ringcrg 15650  ℂfldccnfld 16693
This theorem is referenced by:  plypf1  20121  taylpfval  20271  jensen  20817  amgmlem  20818  lgseisenlem4  21126  esumpfinval  24455  esumpfinvalf  24456  esumpcvgval  24458  esumcvg  24466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058  ax-addf 9059  ax-mulf 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-rp 10603  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-plusg 13532  df-mulr 13533  df-starv 13534  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-0g 13717  df-gsum 13718  df-mnd 14680  df-grp 14802  df-minusg 14803  df-cntz 15106  df-cmn 15404  df-abl 15405  df-mgp 15639  df-rng 15653  df-cring 15654  df-ur 15655  df-cnfld 16694
  Copyright terms: Public domain W3C validator