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Theorem gsumfsum 18804
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 4475 . . . . . . 7  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  ( k  e.  (/)  |->  B ) )
2 mpt0 5691 . . . . . . 7  |-  ( k  e.  (/)  |->  B )  =  (/)
31, 2syl6eq 2459 . . . . . 6  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  (/) )
43oveq2d 6294 . . . . 5  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (fld 
gsumg  (/) ) )
5 cnfld0 18762 . . . . . . 7  |-  0  =  ( 0g ` fld )
65gsum0 16229 . . . . . 6  |-  (fld  gsumg  (/) )  =  0
7 sum0 13692 . . . . . 6  |-  sum_ k  e.  (/)  B  =  0
86, 7eqtr4i 2434 . . . . 5  |-  (fld  gsumg  (/) )  =  sum_ k  e.  (/)  B
94, 8syl6eq 2459 . . . 4  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  (/)  B )
10 sumeq1 13660 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
119, 10eqtr4d 2446 . . 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 18744 . . . . . . 7  |-  CC  =  ( Base ` fld )
14 cnfldadd 18745 . . . . . . 7  |-  +  =  ( +g  ` fld )
15 eqid 2402 . . . . . . 7  |-  (Cntz ` fld )  =  (Cntz ` fld )
16 cnring 18760 . . . . . . . 8  |-fld  e.  Ring
17 ringmnd 17527 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
1816, 17mp1i 13 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->fld  e.  Mnd )
19 gsumfsum.1 . . . . . . . 8  |-  ( ph  ->  A  e.  Fin )
2019adantr 463 . . . . . . 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 2402 . . . . . . . . 9  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
2321, 22fmptd 6033 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
2423adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
k  e.  A  |->  B ) : A --> CC )
25 ringcmn 17549 . . . . . . . . 9  |-  (fld  e.  Ring  ->fld  e. CMnd )
2616, 25mp1i 13 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->fld  e. CMnd )
2713, 15, 26, 24cntzcmnf 17175 . . . . . . 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 758 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
30 f1of1 5798 . . . . . . . 8  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-1-1-> A )
3129, 30syl 17 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-> A )
32 suppssdm 6915 . . . . . . . . 9  |-  ( ( k  e.  A  |->  B ) supp  0 )  C_  dom  ( k  e.  A  |->  B )
33 fdm 5718 . . . . . . . . . 10  |-  ( ( k  e.  A  |->  B ) : A --> CC  ->  dom  ( k  e.  A  |->  B )  =  A )
3424, 33syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  dom  ( k  e.  A  |->  B )  =  A )
3532, 34syl5sseq 3490 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B ) supp  0 ) 
C_  A )
36 f1ofo 5806 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-onto-> A )
37 forn 5781 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -onto-> A  ->  ran  f  =  A
)
3829, 36, 373syl 18 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ran  f  =  A )
3935, 38sseqtr4d 3479 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B ) supp  0 ) 
C_  ran  f )
40 eqid 2402 . . . . . . 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 17235 . . . . . 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 13680 . . . . . . 7  |-  sum_ x  e.  A  ( (
k  e.  A  |->  B ) `  x )  =  sum_ k  e.  A  B
43 fveq2 5849 . . . . . . . 8  |-  ( x  =  ( f `  n )  ->  (
( k  e.  A  |->  B ) `  x
)  =  ( ( k  e.  A  |->  B ) `  ( f `
 n ) ) )
4424ffvelrnda 6009 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  A )  ->  ( ( k  e.  A  |->  B ) `  x )  e.  CC )
45 f1of 5799 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) ) --> A )
4629, 45syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) --> A )
47 fvco3 5926 . . . . . . . . 9  |-  ( ( f : ( 1 ... ( # `  A
) ) --> A  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  n )  =  ( ( k  e.  A  |->  B ) `  (
f `  n )
) )
4846, 47sylan 469 . . . . . . . 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 13691 . . . . . . 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 2457 . . . . . 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 2446 . . . . 5  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
5251expr 613 . . . 4  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
5352exlimdv 1745 . . 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 601 . 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 13681 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
5619, 55syl 17 . 2  |-  ( ph  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
5712, 54, 56mpjaod 379 1  |-  ( ph  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   (/)c0 3738    |-> cmpt 4453   dom cdm 4823   ran crn 4824    o. ccom 4827   -->wf 5565   -1-1->wf1 5566   -onto->wfo 5567   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278   supp csupp 6902   Fincfn 7554   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525   NNcn 10576   ...cfz 11726    seqcseq 12151   #chash 12452   sum_csu 13657    gsumg cgsu 15055   Mndcmnd 16243  Cntzccntz 16677  CMndccmn 17122   Ringcrg 17518  ℂfldccnfld 18740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-plusg 14922  df-mulr 14923  df-starv 14924  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-0g 15056  df-gsum 15057  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-cnfld 18741
This theorem is referenced by:  regsumsupp  18956  plypf1  22901  taylpfval  23052  jensen  23644  amgmlem  23645  lgseisenlem4  24008  regsumfsum  28224  esumpfinval  28522  esumpfinvalf  28523  esumpcvgval  28525  esumcvg  28533  aacllem  38860
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