Theorem fsumconst 14522

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.

Step	Hyp	Ref	Expression
1		mul02 10214	. . . . 5 |- ( B e. CC -> ( 0 x. B ) = 0 )
2	1	adantl 482	. . . 4 |- ( ( A e. Fin /\ B e. CC ) -> ( 0 x. B ) = 0 )
3	2	eqcomd 2628	. . 3 |- ( ( A e. Fin /\ B e. CC ) -> 0 = ( 0 x. B ) )
4		sumeq1 14419	. . . . 5 |- ( A = (/) -> sum_ k e. A B = sum_ k e. (/) B )
5		sum0 14452	. . . . 5 |- sum_ k e. (/) B = 0
6	4, 5	syl6eq 2672	. . . 4 |- ( A = (/) -> sum_ k e. A B = 0 )
7		fveq2 6191	. . . . . 6 |- ( A = (/) -> ( # ` A ) = ( # ` (/) ) )
8		hash0 13158	. . . . . 6 |- ( # ` (/) ) = 0
9	7, 8	syl6eq 2672	. . . . 5 |- ( A = (/) -> ( # ` A ) = 0 )
10	9	oveq1d 6665	. . . 4 |- ( A = (/) -> ( ( # ` A ) x. B ) = ( 0 x. B ) )
11	6, 10	eqeq12d 2637	. . 3 |- ( A = (/) -> ( sum_ k e. A B = ( ( # ` A ) x. B ) <-> 0 = ( 0 x. B ) ) )
12	3, 11	syl5ibrcom 237	. 2 |- ( ( A e. Fin /\ B e. CC ) -> ( A = (/) -> sum_ k e. A B = ( ( # ` A ) x. B ) ) )
13		eqidd 2623	. . . . . . 7 |- ( k = ( f ` n ) -> B = B )
14		simprl 794	. . . . . . 7 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( # ` A ) e. NN )
15		simprr 796	. . . . . . 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 799	. . . . . . 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 792	. . . . . . . 8 |- ( ( ( A e. Fin /\ B e. CC ) /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> B e. CC )
18		elfznn 12370	. . . . . . . 8 |- ( n e. ( 1 ... ( # ` A ) ) -> n e. NN )
19		fvconst2g 6467	. . . . . . . 8 |- ( ( B e. CC /\ n e. NN ) -> ( ( NN X. { B } ) ` n ) = B )
20	17, 18, 19	syl2an 494	. . . . . . 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 )
21	13, 14, 15, 16, 20	fsum 14451	. . . . . 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 12857	. . . . . . 7 |- ( ( B e. CC /\ ( # ` A ) e. NN ) -> ( seq 1 ( + , ( NN X. { B } ) ) ` ( # ` A ) ) = ( ( # ` A ) x. B ) )
23	22	ad2ant2lr 784	. . . . . 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 ) )
24	21, 23	eqtrd 2656	. . . . 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 ) )
25	24	expr 643	. . . 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 ) ) )
26	25	exlimdv 1861	. . 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 ) ) )
27	26	expimpd 629	. 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 14441	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
29	28	adantr 481	. 2 |- ( ( A e. Fin /\ B e. CC ) -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
30	12, 27, 29	mpjaod 396	1 |- ( ( A e. Fin /\ B e. CC ) -> sum_ k e. A B = ( ( # ` A ) x. B ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   {csn 4177   X. cxp 5112   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   ...cfz 12326   seqcseq 12801   #chash 13117   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  fsumdifsnconst  14523  o1fsum  14545  hashiun  14554  hash2iun1dif1  14556  climcndslem1  14581  climcndslem2  14582  harmonic  14591  mertenslem1  14616  sumhash  15600  cshwshashnsame  15810  lagsubg2  17655  sylow2a  18034  lebnumlem3  22762  uniioombllem4  23354  birthdaylem2  24679  basellem8  24814  0sgm  24870  musum  24917  chtleppi  24935  vmasum  24941  logfac2  24942  chpval2  24943  chpchtsum  24944  chpub  24945  logfaclbnd  24947  dchrsum2  24993  sumdchr2  24995  lgsquadlem1  25105  chebbnd1lem1  25158  chtppilimlem1  25162  dchrmusum2  25183  dchrisum0flblem1  25197  rpvmasum2  25201  dchrisum0lem2a  25206  mudivsum  25219  mulogsumlem  25220  selberglem2  25235  pntlemj  25292  rusgrnumwwlks  26869  fusgrhashclwwlkn  26956  fusgreghash2wsp  27202  numclwwlk6  27248  reprlt  30697  hashreprin  30698  reprgt  30699  hgt750lema  30735  rrndstprj2  33630  stoweidlem11  40228  stoweidlem26  40243  stoweidlem38  40255  dirkertrigeq  40318  fourierdlem73  40396  etransclem32  40483  rrndistlt  40510  sge0rpcpnf  40638  hoiqssbllem2  40837  nn0mulfsum  42418  amgmlemALT  42549