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Theorem fsumconst 12575
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 9246 . . . . 5  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
21adantl 454 . . . 4  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( 0  x.  B
)  =  0 )
32eqcomd 2443 . . 3  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  0  =  ( 0  x.  B ) )
4 sumeq1 12485 . . . . 5  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  =  sum_ k  e.  (/)  B )
5 sum0 12517 . . . . 5  |-  sum_ k  e.  (/)  B  =  0
64, 5syl6eq 2486 . . . 4  |-  ( A  =  (/)  ->  sum_ k  e.  A  B  = 
0 )
7 fveq2 5730 . . . . . 6  |-  ( A  =  (/)  ->  ( # `  A )  =  (
# `  (/) ) )
8 hash0 11648 . . . . . 6  |-  ( # `  (/) )  =  0
97, 8syl6eq 2486 . . . . 5  |-  ( A  =  (/)  ->  ( # `  A )  =  0 )
109oveq1d 6098 . . . 4  |-  ( A  =  (/)  ->  ( (
# `  A )  x.  B )  =  ( 0  x.  B ) )
116, 10eqeq12d 2452 . . 3  |-  ( A  =  (/)  ->  ( sum_ k  e.  A  B  =  ( ( # `  A )  x.  B
)  <->  0  =  ( 0  x.  B ) ) )
123, 11syl5ibrcom 215 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  -> 
sum_ k  e.  A  B  =  ( ( # `
 A )  x.  B ) ) )
13 eqidd 2439 . . . . . . 7  |-  ( k  =  ( f `  n )  ->  B  =  B )
14 simprl 734 . . . . . . 7  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  -> 
( # `  A )  e.  NN )
15 simprr 735 . . . . . . 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 737 . . . . . . 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 733 . . . . . . . 8  |-  ( ( ( A  e.  Fin  /\  B  e.  CC )  /\  ( ( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  ->  B  e.  CC )
18 elfznn 11082 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( # `  A
) )  ->  n  e.  NN )
19 fvconst2g 5947 . . . . . . . 8  |-  ( ( B  e.  CC  /\  n  e.  NN )  ->  ( ( NN  X.  { B } ) `  n )  =  B )
2017, 18, 19syl2an 465 . . . . . . 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 12516 . . . . . 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 11381 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( # `  A )  e.  NN )  -> 
(  seq  1 (  +  ,  ( NN 
X.  { B }
) ) `  ( # `
 A ) )  =  ( ( # `  A )  x.  B
) )
2322ad2ant2lr 730 . . . . . 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 2470 . . . . 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 600 . . . 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 1647 . . 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 588 . 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 12506 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
2928adantr 453 . 2  |-  ( ( A  e.  Fin  /\  B  e.  CC )  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3012, 27, 29mpjaod 372 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 359    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   (/)c0 3630   {csn 3816    X. cxp 4878   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Fincfn 7111   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997   NNcn 10002   ...cfz 11045    seq cseq 11325   #chash 11620   sum_csu 12481
This theorem is referenced by:  o1fsum  12594  hashiun  12603  climcndslem1  12631  climcndslem2  12632  harmonic  12640  mertenslem1  12663  sumhash  13267  lagsubg2  15003  sylow2a  15255  lebnumlem3  18990  uniioombllem4  19480  birthdaylem2  20793  basellem8  20872  0sgm  20929  musum  20978  chtleppi  20996  vmasum  21002  logfac2  21003  chpval2  21004  chpchtsum  21005  chpub  21006  logfaclbnd  21008  dchrsum2  21054  sumdchr2  21056  lgsquadlem1  21140  chebbnd1lem1  21165  chtppilimlem1  21169  dchrmusum2  21190  dchrisum0flblem1  21204  rpvmasum2  21208  dchrisum0lem2a  21213  mudivsum  21226  mulogsumlem  21227  selberglem2  21242  pntlemj  21299  rrndstprj2  26542  stoweidlem11  27738  stoweidlem26  27753  stoweidlem38  27765  cshwssizensame  28311  frghash2spot  28514  usgreghash2spotv  28517  usgreghash2spot  28520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482
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