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Theorem fsumconst 12528
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 9200 . . . . 5  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
21adantl 453 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
32eqcomd 2409 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  0  =  ( 0  x.  B ) )
4 sumeq1 12438 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
5 sum0 12470 . . . . 5  |-  sum_ k  e.  (/)  B  =  0
64, 5syl6eq 2452 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  = 
0 )
7 fveq2 5687 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
8 hash0 11601 . . . . . 6  |-  ( # `  (/) )  =  0
97, 8syl6eq 2452 . . . . 5  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
109oveq1d 6055 . . . 4  |-  ( A  =  (/)  ->  ( (
# `  A )  x.  B )  =  ( 0  x.  B ) )
116, 10eqeq12d 2418 . . 3  |-  ( A  =  (/)  ->  ( sum_ k  e.  A  B  =  ( ( # `  A )  x.  B
)  <->  0  =  ( 0  x.  B ) ) )
123, 11syl5ibrcom 214 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) ) )
13 eqidd 2405 . . . . . . 7  |-  ( k  =  ( f `  n )  ->  B  =  B )
14 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( # `  A )  e.  NN )
15 simprr 734 . . . . . . 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 736 . . . . . . 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 732 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  B  e.  CC )
18 elfznn 11036 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
19 fvconst2g 5904 . . . . . . . 8  |-  ( ( B  e.  CC  /\  n  e.  NN )  ->  ( ( NN  X.  { B } ) `  n )  =  B )
2017, 18, 19syl2an 464 . . . . . . 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 12469 . . . . . 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 11334 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( # `  A )  e.  NN )  -> 
(  seq  1 (  +  ,  ( NN 
X.  { B }
) ) `  ( # `
 A ) )  =  ( ( # `  A )  x.  B
) )
2322ad2ant2lr 729 . . . . . 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 2436 . . . . 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 599 . . . 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 1643 . . 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 587 . 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 12459 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
2928adantr 452 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3012, 27, 29mpjaod 371 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 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   (/)c0 3588   {csn 3774    X. cxp 4835   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   NNcn 9956   ...cfz 10999    seq cseq 11278   #chash 11573   sum_csu 12434
This theorem is referenced by:  o1fsum  12547  hashiun  12556  climcndslem1  12584  climcndslem2  12585  harmonic  12593  mertenslem1  12616  sumhash  13220  lagsubg2  14956  sylow2a  15208  lebnumlem3  18941  uniioombllem4  19431  birthdaylem2  20744  basellem8  20823  0sgm  20880  musum  20929  chtleppi  20947  vmasum  20953  logfac2  20954  chpval2  20955  chpchtsum  20956  chpub  20957  logfaclbnd  20959  dchrsum2  21005  sumdchr2  21007  lgsquadlem1  21091  chebbnd1lem1  21116  chtppilimlem1  21120  dchrmusum2  21141  dchrisum0flblem1  21155  rpvmasum2  21159  dchrisum0lem2a  21164  mudivsum  21177  mulogsumlem  21178  selberglem2  21193  pntlemj  21250  rrndstprj2  26430  stoweidlem11  27627  stoweidlem26  27642  stoweidlem38  27654  frghash2spot  28166  usgreghash2spotv  28169  usgreghash2spot  28172
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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