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Theorem gsumfsum 17722
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 4360 . . . . . . 7  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  ( k  e.  (/)  |->  B ) )
2 mpt0 5526 . . . . . . 7  |-  ( k  e.  (/)  |->  B )  =  (/)
31, 2syl6eq 2481 . . . . . 6  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  (/) )
43oveq2d 6096 . . . . 5  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (fld 
gsumg  (/) ) )
5 cnfld0 17683 . . . . . . 7  |-  0  =  ( 0g ` fld )
65gsum0 15489 . . . . . 6  |-  (fld  gsumg  (/) )  =  0
7 sum0 13181 . . . . . 6  |-  sum_ k  e.  (/)  B  =  0
86, 7eqtr4i 2456 . . . . 5  |-  (fld  gsumg  (/) )  =  sum_ k  e.  (/)  B
94, 8syl6eq 2481 . . . 4  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  (/)  B )
10 sumeq1 13149 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
119, 10eqtr4d 2468 . . 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 17665 . . . . . . 7  |-  CC  =  ( Base ` fld )
14 cnfldadd 17666 . . . . . . 7  |-  +  =  ( +g  ` fld )
15 eqid 2433 . . . . . . 7  |-  (Cntz ` fld )  =  (Cntz ` fld )
16 cnrng 17681 . . . . . . . 8  |-fld  e.  Ring
17 rngmnd 16589 . . . . . . . 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 462 . . . . . . 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 2433 . . . . . . . . 9  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
2321, 22fmptd 5855 . . . . . . . 8  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> CC )
2423adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
k  e.  A  |->  B ) : A --> CC )
25 rngcmn 16610 . . . . . . . . 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 16306 . . . . . . 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 748 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( # `
 A )  e.  NN )
29 simprr 749 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
30 f1of1 5628 . . . . . . . 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 6692 . . . . . . . . 9  |-  ( ( k  e.  A  |->  B ) supp  0 )  C_  dom  ( k  e.  A  |->  B )
33 fdm 5551 . . . . . . . . . 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 3392 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B ) supp  0 ) 
C_  A )
36 f1ofo 5636 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-onto-> A )
37 forn 5611 . . . . . . . . 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 3381 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B ) supp  0 ) 
C_  ran  f )
40 eqid 2433 . . . . . . 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 16364 . . . . . 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 13169 . . . . . . 7  |-  sum_ x  e.  A  ( (
k  e.  A  |->  B ) `  x )  =  sum_ k  e.  A  B
43 fveq2 5679 . . . . . . . 8  |-  ( x  =  ( f `  n )  ->  (
( k  e.  A  |->  B ) `  x
)  =  ( ( k  e.  A  |->  B ) `  ( f `
 n ) ) )
4424ffvelrnda 5831 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  A )  ->  ( ( k  e.  A  |->  B ) `  x )  e.  CC )
45 f1of 5629 . . . . . . . . . 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 5756 . . . . . . . . 9  |-  ( ( f : ( 1 ... ( # `  A
) ) --> A  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  n )  =  ( ( k  e.  A  |->  B ) `  (
f `  n )
) )
4846, 47sylan 468 . . . . . . . 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 13180 . . . . . . 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 2479 . . . . . 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 2468 . . . . 5  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B )
5251expr 610 . . . 4  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  (fld 
gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  A  B ) )
5352exlimdv 1689 . . 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 598 . 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 13170 . . 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 1362   E.wex 1589    e. wcel 1755   (/)c0 3625    e. cmpt 4338   dom cdm 4827   ran crn 4828    o. ccom 4831   -->wf 5402   -1-1->wf1 5403   -onto->wfo 5404   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   supp csupp 6679   Fincfn 7298   CCcc 9267   0cc0 9269   1c1 9270    + caddc 9272   NNcn 10309   ...cfz 11423    seqcseq 11789   #chash 12086   sum_csu 13146    gsumg cgsu 14361   Mndcmnd 15391  Cntzccntz 15812  CMndccmn 16256   Ringcrg 16576  ℂfldccnfld 17661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-rp 10979  df-fz 11424  df-fzo 11532  df-seq 11790  df-exp 11849  df-hash 12087  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-clim 12949  df-sum 13147  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-plusg 14233  df-mulr 14234  df-starv 14235  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-0g 14362  df-gsum 14363  df-mnd 15397  df-grp 15524  df-minusg 15525  df-cntz 15814  df-cmn 16258  df-abl 16259  df-mgp 16565  df-rng 16579  df-cring 16580  df-ur 16581  df-cnfld 17662
This theorem is referenced by:  regsumsupp  17893  plypf1  21564  taylpfval  21714  jensen  22266  amgmlem  22267  lgseisenlem4  22575  regsumfsum  26102  esumpfinval  26377  esumpfinvalf  26378  esumpcvgval  26380  esumcvg  26388
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