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Theorem gsumfsum 18463
Description: Relate a group sum on ℂfld to a finite sum on the complex numbers. (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 4517 . . . . . . 7  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  ( k  e.  (/)  |->  B ) )
2 mpt0 5698 . . . . . . 7  |-  ( k  e.  (/)  |->  B )  =  (/)
31, 2syl6eq 2500 . . . . . 6  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  (/) )
43oveq2d 6297 . . . . 5  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (fld 
gsumg  (/) ) )
5 cnfld0 18421 . . . . . . 7  |-  0  =  ( 0g ` fld )
65gsum0 15884 . . . . . 6  |-  (fld  gsumg  (/) )  =  0
7 sum0 13525 . . . . . 6  |-  sum_ k  e.  (/)  B  =  0
86, 7eqtr4i 2475 . . . . 5  |-  (fld  gsumg  (/) )  =  sum_ k  e.  (/)  B
94, 8syl6eq 2500 . . . 4  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  (/)  B )
10 sumeq1 13493 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
119, 10eqtr4d 2487 . . 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 18403 . . . . . . 7  |-  CC  =  ( Base ` fld )
14 cnfldadd 18404 . . . . . . 7  |-  +  =  ( +g  ` fld )
15 eqid 2443 . . . . . . 7  |-  (Cntz ` fld )  =  (Cntz ` fld )
16 cnring 18419 . . . . . . . 8  |-fld  e.  Ring
17 ringmnd 17186 . . . . . . . 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 465 . . . . . . 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 2443 . . . . . . . . 9  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
2321, 22fmptd 6040 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
2423adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
k  e.  A  |->  B ) : A --> CC )
25 ringcmn 17208 . . . . . . . . 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 16830 . . . . . . 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 756 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( # `
 A )  e.  NN )
29 simprr 757 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
30 f1of1 5805 . . . . . . . 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 suppssdm 6916 . . . . . . . . 9  |-  ( ( k  e.  A  |->  B ) supp  0 )  C_  dom  ( k  e.  A  |->  B )
33 fdm 5725 . . . . . . . . . 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 3537 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B ) supp  0 ) 
C_  A )
36 f1ofo 5813 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-onto-> A )
37 forn 5788 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -onto-> A  ->  ran  f  =  A
)
3829, 36, 373syl 20 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ran  f  =  A )
3935, 38sseqtr4d 3526 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B ) supp  0 ) 
C_  ran  f )
40 eqid 2443 . . . . . . 7  |-  ( ( ( k  e.  A  |->  B )  o.  f
) supp  0 )  =  ( ( ( k  e.  A  |->  B )  o.  f ) supp  0
)
4113, 5, 14, 15, 18, 20, 24, 27, 28, 31, 39, 40gsumval3 16890 . . . . . 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 13513 . . . . . . 7  |-  sum_ x  e.  A  ( (
k  e.  A  |->  B ) `  x )  =  sum_ k  e.  A  B
43 fveq2 5856 . . . . . . . 8  |-  ( x  =  ( f `  n )  ->  (
( k  e.  A  |->  B ) `  x
)  =  ( ( k  e.  A  |->  B ) `  ( f `
 n ) ) )
4424ffvelrnda 6016 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  A )  ->  ( ( k  e.  A  |->  B ) `  x )  e.  CC )
45 f1of 5806 . . . . . . . . . 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 5935 . . . . . . . . 9  |-  ( ( f : ( 1 ... ( # `  A
) ) --> A  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  n )  =  ( ( k  e.  A  |->  B ) `  (
f `  n )
) )
4846, 47sylan 471 . . . . . . . 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 13524 . . . . . . 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 2498 . . . . . 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 2487 . . . . 5  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
5251expr 615 . . . 4  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
5352exlimdv 1711 . . 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 603 . 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 13514 . . 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 381 1  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   (/)c0 3770    |-> cmpt 4495   dom cdm 4989   ran crn 4990    o. ccom 4993   -->wf 5574   -1-1->wf1 5575   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   supp csupp 6903   Fincfn 7518   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498   NNcn 10543   ...cfz 11683    seqcseq 12089   #chash 12387   sum_csu 13490    gsumg cgsu 14820   Mndcmnd 15898  Cntzccntz 16332  CMndccmn 16777   Ringcrg 17177  ℂfldccnfld 18399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-rp 11232  df-fz 11684  df-fzo 11807  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-sum 13491  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-plusg 14692  df-mulr 14693  df-starv 14694  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-0g 14821  df-gsum 14822  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-cntz 16334  df-cmn 16779  df-abl 16780  df-mgp 17121  df-ur 17133  df-ring 17179  df-cring 17180  df-cnfld 18400
This theorem is referenced by:  regsumsupp  18636  plypf1  22587  taylpfval  22738  jensen  23296  amgmlem  23297  lgseisenlem4  23605  regsumfsum  27750  esumpfinval  28059  esumpfinvalf  28060  esumpcvgval  28062  esumcvg  28070
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