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Theorem fsumconst 13240
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 9535 . . . . 5  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
21adantl 463 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
32eqcomd 2438 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  0  =  ( 0  x.  B ) )
4 sumeq1 13150 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
5 sum0 13182 . . . . 5  |-  sum_ k  e.  (/)  B  =  0
64, 5syl6eq 2481 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  = 
0 )
7 fveq2 5679 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
8 hash0 12119 . . . . . 6  |-  ( # `  (/) )  =  0
97, 8syl6eq 2481 . . . . 5  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
109oveq1d 6095 . . . 4  |-  ( A  =  (/)  ->  ( (
# `  A )  x.  B )  =  ( 0  x.  B ) )
116, 10eqeq12d 2447 . . 3  |-  ( A  =  (/)  ->  ( sum_ k  e.  A  B  =  ( ( # `  A )  x.  B
)  <->  0  =  ( 0  x.  B ) ) )
123, 11syl5ibrcom 222 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) ) )
13 eqidd 2434 . . . . . . 7  |-  ( k  =  ( f `  n )  ->  B  =  B )
14 simprl 748 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( # `  A )  e.  NN )
15 simprr 749 . . . . . . 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 751 . . . . . . 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 747 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  B  e.  CC )
18 elfznn 11465 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
19 fvconst2g 5918 . . . . . . . 8  |-  ( ( B  e.  CC  /\  n  e.  NN )  ->  ( ( NN  X.  { B } ) `  n )  =  B )
2017, 18, 19syl2an 474 . . . . . . 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 13181 . . . . . 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 11846 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( # `  A )  e.  NN )  -> 
(  seq 1 (  +  ,  ( NN  X.  { B } ) ) `
 ( # `  A
) )  =  ( ( # `  A
)  x.  B ) )
2322ad2ant2lr 740 . . . . . 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 2465 . . . . 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 610 . . . 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 1689 . . 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 598 . 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 13171 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
2928adantr 462 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3012, 27, 29mpjaod 381 1  |-  ( ( A  e.  Fin  /\  B  e.  CC )  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  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   {csn 3865    X. cxp 4825   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275   NNcn 10310   ...cfz 11424    seqcseq 11790   #chash 12087   sum_csu 13147
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 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  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-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 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148
This theorem is referenced by:  o1fsum  13259  hashiun  13268  climcndslem1  13295  climcndslem2  13296  harmonic  13304  mertenslem1  13327  sumhash  13941  cshwshashnsame  14113  lagsubg2  15722  sylow2a  16098  lebnumlem3  20377  uniioombllem4  20908  birthdaylem2  22231  basellem8  22310  0sgm  22367  musum  22416  chtleppi  22434  vmasum  22440  logfac2  22441  chpval2  22442  chpchtsum  22443  chpub  22444  logfaclbnd  22446  dchrsum2  22492  sumdchr2  22494  lgsquadlem1  22578  chebbnd1lem1  22603  chtppilimlem1  22607  dchrmusum2  22628  dchrisum0flblem1  22642  rpvmasum2  22646  dchrisum0lem2a  22651  mudivsum  22664  mulogsumlem  22665  selberglem2  22680  pntlemj  22737  rrndstprj2  28574  stoweidlem11  29652  stoweidlem26  29667  stoweidlem38  29679  hashclwwlkn  30356  rusgranumwlks  30420  frghash2spot  30502  usgreghash2spotv  30505  usgreghash2spot  30508  numclwwlk6  30552
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