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Theorem fsumconst 12252
Description: The sum of constant terms ( k is not free in  A). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Assertion
Ref Expression
fsumconst  |-  ( ( A  e.  Fin  /\  B  e.  CC )  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) )
Distinct variable groups:    A, k    B, k

Proof of Theorem fsumconst
Dummy variables  f  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mul02 8990 . . . . 5  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
21adantl 452 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
32eqcomd 2288 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  0  =  ( 0  x.  B ) )
4 sumeq1 12162 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
5 sum0 12194 . . . . 5  |-  sum_ k  e.  (/)  B  =  0
64, 5syl6eq 2331 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  = 
0 )
7 fveq2 5525 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
8 hash0 11355 . . . . . 6  |-  ( # `  (/) )  =  0
97, 8syl6eq 2331 . . . . 5  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
109oveq1d 5873 . . . 4  |-  ( A  =  (/)  ->  ( (
# `  A )  x.  B )  =  ( 0  x.  B ) )
116, 10eqeq12d 2297 . . 3  |-  ( A  =  (/)  ->  ( sum_ k  e.  A  B  =  ( ( # `  A )  x.  B
)  <->  0  =  ( 0  x.  B ) ) )
123, 11syl5ibrcom 213 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) ) )
13 eqidd 2284 . . . . . . 7  |-  ( k  =  ( f `  n )  ->  B  =  B )
14 simprl 732 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( # `  A )  e.  NN )
15 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
16 simpllr 735 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  CC )  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  k  e.  A )  ->  B  e.  CC )
17 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  B  e.  CC )
18 elfznn 10819 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
19 fvconst2g 5727 . . . . . . . 8  |-  ( ( B  e.  CC  /\  n  e.  NN )  ->  ( ( NN  X.  { B } ) `  n )  =  B )
2017, 18, 19syl2an 463 . . . . . . 7  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  CC )  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  n  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( NN  X.  { B } ) `  n )  =  B )
2113, 14, 15, 16, 20fsum 12193 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  B  =  (  seq  1
(  +  ,  ( NN  X.  { B } ) ) `  ( # `  A ) ) )
22 ser1const 11102 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( # `  A )  e.  NN )  -> 
(  seq  1 (  +  ,  ( NN 
X.  { B }
) ) `  ( # `
 A ) )  =  ( ( # `  A )  x.  B
) )
2322ad2ant2lr 728 . . . . . 6  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
(  seq  1 (  +  ,  ( NN 
X.  { B }
) ) `  ( # `
 A ) )  =  ( ( # `  A )  x.  B
) )
2421, 23eqtrd 2315 . . . . 5  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  B  =  ( ( # `  A )  x.  B
) )
2524expr 598 . . . 4  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) ) )
2625exlimdv 1664 . . 3  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( # `  A
)  e.  NN )  ->  ( E. f 
f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  B  =  ( ( # `  A
)  x.  B ) ) )
2726expimpd 586 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( ( ( # `  A )  e.  NN  /\ 
E. f  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A  B  =  ( ( # `  A
)  x.  B ) ) )
28 fz1f1o 12183 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
2928adantr 451 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3012, 27, 29mpjaod 370 1  |-  ( ( A  e.  Fin  /\  B  e.  CC )  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   (/)c0 3455   {csn 3640    X. cxp 4687   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   ...cfz 10782    seq cseq 11046   #chash 11337   sum_csu 12158
This theorem is referenced by:  o1fsum  12271  hashiun  12280  climcndslem1  12308  climcndslem2  12309  harmonic  12317  mertenslem1  12340  sumhash  12944  lagsubg2  14678  sylow2a  14930  lebnumlem3  18461  uniioombllem4  18941  birthdaylem2  20247  basellem8  20325  0sgm  20382  musum  20431  chtleppi  20449  vmasum  20455  logfac2  20456  chpval2  20457  chpchtsum  20458  chpub  20459  logfaclbnd  20461  dchrsum2  20507  sumdchr2  20509  lgsquadlem1  20593  chebbnd1lem1  20618  chtppilimlem1  20622  dchrmusum2  20643  dchrisum0flblem1  20657  rpvmasum2  20661  dchrisum0lem2a  20666  mudivsum  20679  mulogsumlem  20680  selberglem2  20695  pntlemj  20752  rrndstprj2  26555  stoweidlem11  27760  stoweidlem26  27775  stoweidlem38  27787  stoweidlem44  27793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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