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Theorem gsumfsum 16690
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 4231 . . . . . . 7  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  ( k  e.  (/)  |->  B ) )
2 mpt0 5513 . . . . . . 7  |-  ( k  e.  (/)  |->  B )  =  (/)
31, 2syl6eq 2436 . . . . . 6  |-  ( A  =  (/)  ->  ( k  e.  A  |->  B )  =  (/) )
43oveq2d 6037 . . . . 5  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  =  (fld 
gsumg  (/) ) )
5 cnfld0 16649 . . . . . . 7  |-  0  =  ( 0g ` fld )
65gsum0 14708 . . . . . 6  |-  (fld  gsumg  (/) )  =  0
7 sum0 12443 . . . . . 6  |-  sum_ k  e.  (/)  B  =  0
86, 7eqtr4i 2411 . . . . 5  |-  (fld  gsumg  (/) )  =  sum_ k  e.  (/)  B
94, 8syl6eq 2436 . . . 4  |-  ( A  =  (/)  ->  (fld  gsumg  ( k  e.  A  |->  B ) )  = 
sum_ k  e.  (/)  B )
10 sumeq1 12411 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
119, 10eqtr4d 2423 . . 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 16631 . . . . . . 7  |-  CC  =  ( Base ` fld )
14 cnfldadd 16632 . . . . . . 7  |-  +  =  ( +g  ` fld )
15 eqid 2388 . . . . . . 7  |-  (Cntz ` fld )  =  (Cntz ` fld )
16 cnrng 16647 . . . . . . . 8  |-fld  e.  Ring
17 rngmnd 15601 . . . . . . . 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 2388 . . . . . . . . 9  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
2321, 22fmptd 5833 . . . . . . . 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 15622 . . . . . . . . 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 15443 . . . . . . 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 5614 . . . . . . . 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 5165 . . . . . . . . 9  |-  ( `' ( k  e.  A  |->  B ) " ( _V  \  { 0 } ) )  C_  dom  ( k  e.  A  |->  B )
33 fdm 5536 . . . . . . . . . 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 3340 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( `' ( k  e.  A  |->  B ) "
( _V  \  {
0 } ) ) 
C_  A )
36 f1ofo 5622 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) )
-onto-> A )
37 forn 5597 . . . . . . . . 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 3329 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( `' ( k  e.  A  |->  B ) "
( _V  \  {
0 } ) ) 
C_  ran  f )
40 eqid 2388 . . . . . . 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 15442 . . . . . 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 12431 . . . . . . 7  |-  sum_ x  e.  A  ( (
k  e.  A  |->  B ) `  x )  =  sum_ k  e.  A  B
43 fveq2 5669 . . . . . . . 8  |-  ( x  =  ( f `  n )  ->  (
( k  e.  A  |->  B ) `  x
)  =  ( ( k  e.  A  |->  B ) `  ( f `
 n ) ) )
4424ffvelrnda 5810 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  A )  ->  ( ( k  e.  A  |->  B ) `  x )  e.  CC )
45 f1of 5615 . . . . . . . . . 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 5740 . . . . . . . . 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 12442 . . . . . . 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 2434 . . . . . 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 2423 . . . . 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 1643 . . 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 12432 . . 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 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900    \ cdif 3261   (/)c0 3572   {csn 3758    e. cmpt 4208   `'ccnv 4818   dom cdm 4819   ran crn 4820   "cima 4822    o. ccom 4823   -->wf 5391   -1-1->wf1 5392   -onto->wfo 5393   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   Fincfn 7046   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927   NNcn 9933   ...cfz 10976    seq cseq 11251   #chash 11546   sum_csu 12407    gsumg cgsu 13652   Mndcmnd 14612  Cntzccntz 15042  CMndccmn 15340   Ringcrg 15588  ℂfldccnfld 16627
This theorem is referenced by:  plypf1  19999  taylpfval  20149  jensen  20695  amgmlem  20696  lgseisenlem4  21004  esumpfinval  24262  esumpfinvalf  24263  esumpcvgval  24265  esumcvg  24273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-rp 10546  df-fz 10977  df-fzo 11067  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-sum 12408  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-plusg 13470  df-mulr 13471  df-starv 13472  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-0g 13655  df-gsum 13656  df-mnd 14618  df-grp 14740  df-minusg 14741  df-cntz 15044  df-cmn 15342  df-abl 15343  df-mgp 15577  df-rng 15591  df-cring 15592  df-ur 15593  df-cnfld 16628
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