Theorem gsumfsum 19813

Description: Relate a group sum on ℂ_fld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)

Hypotheses

Ref	Expression
gsumfsum.1	|- ( ph -> A e. Fin )
gsumfsum.2	|- ( ( ph /\ k e. A ) -> B e. CC )

Assertion

Ref	Expression
gsumfsum	|- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )

Distinct variable groups:   A, k   ph, k

Allowed substitution hint:   B( k)

Proof of Theorem gsumfsum

Dummy variables f n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq1 4737	. . . . . . 7 |- ( A = (/) -> ( k e. A |-> B ) = ( k e. (/) |-> B ) )
2		mpt0 6021	. . . . . . 7 |- ( k e. (/) |-> B ) = (/)
3	1, 2	syl6eq 2672	. . . . . 6 |- ( A = (/) -> ( k e. A |-> B ) = (/) )
4	3	oveq2d 6666	. . . . 5 |- ( A = (/) -> (ℂfld gsumg ( k e. A |-> B ) ) = (ℂfld gsumg (/) ) )
5		cnfld0 19770	. . . . . . 7 |- 0 = ( 0g ` ℂfld )
6	5	gsum0 17278	. . . . . 6 |- (ℂfld gsumg (/) ) = 0
7		sum0 14452	. . . . . 6 |- sum_ k e. (/) B = 0
8	6, 7	eqtr4i 2647	. . . . 5 |- (ℂfld gsumg (/) ) = sum_ k e. (/) B
9	4, 8	syl6eq 2672	. . . 4 |- ( A = (/) -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. (/) B )
10		sumeq1 14419	. . . 4 |- ( A = (/) -> sum_ k e. A B = sum_ k e. (/) B )
11	9, 10	eqtr4d 2659	. . 3 |- ( A = (/) -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
12	11	a1i 11	. 2 |- ( ph -> ( A = (/) -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B ) )
13		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
14		cnfldadd 19751	. . . . . . 7 |- + = ( +g ` ℂfld )
15		eqid 2622	. . . . . . 7 |- (Cntz ` ℂfld ) = (Cntz ` ℂfld )
16		cnring 19768	. . . . . . . 8 |- ℂfld e. Ring
17		ringmnd 18556	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
18	16, 17	mp1i 13	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ℂfld e. Mnd )
19		gsumfsum.1	. . . . . . . 8 |- ( ph -> A e. Fin )
20	19	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> A e. Fin )
21		gsumfsum.2	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> B e. CC )
22		eqid 2622	. . . . . . . . 9 |- ( k e. A |-> B ) = ( k e. A |-> B )
23	21, 22	fmptd 6385	. . . . . . . 8 |- ( ph -> ( k e. A |-> B ) : A --> CC )
24	23	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( k e. A |-> B ) : A --> CC )
25		ringcmn 18581	. . . . . . . . 9 |- (ℂfld e. Ring -> ℂfld e. CMnd )
26	16, 25	mp1i 13	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ℂfld e. CMnd )
27	13, 15, 26, 24	cntzcmnf 18248	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ran ( k e. A |-> B ) C_ ( (Cntz ` ℂfld ) ` ran ( k e. A |-> B ) ) )
28		simprl 794	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( # ` A ) e. NN )
29		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> f : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
30		f1of1 6136	. . . . . . . 8 |- ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> f : ( 1 ... ( # ` A ) ) -1-1-> A )
31	29, 30	syl 17	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> f : ( 1 ... ( # ` A ) ) -1-1-> A )
32		suppssdm 7308	. . . . . . . . 9 |- ( ( k e. A |-> B ) supp 0 ) C_ dom ( k e. A |-> B )
33		fdm 6051	. . . . . . . . . 10 |- ( ( k e. A |-> B ) : A --> CC -> dom ( k e. A |-> B ) = A )
34	24, 33	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> dom ( k e. A |-> B ) = A )
35	32, 34	syl5sseq 3653	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( ( k e. A |-> B ) supp 0 ) C_ A )
36		f1ofo 6144	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> f : ( 1 ... ( # ` A ) ) -onto-> A )
37		forn 6118	. . . . . . . . 9 |- ( f : ( 1 ... ( # ` A ) ) -onto-> A -> ran f = A )
38	29, 36, 37	3syl 18	. . . . . . . 8 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ran f = A )
39	35, 38	sseqtr4d 3642	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> ( ( k e. A |-> B ) supp 0 ) C_ ran f )
40		eqid 2622	. . . . . . 7 |- ( ( ( k e. A |-> B ) o. f ) supp 0 ) = ( ( ( k e. A |-> B ) o. f ) supp 0 )
41	13, 5, 14, 15, 18, 20, 24, 27, 28, 31, 39, 40	gsumval3 18308	. . . . . 6 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> (ℂfld gsumg ( k e. A |-> B ) ) = ( seq 1 ( + , ( ( k e. A |-> B ) o. f ) ) ` ( # ` A ) ) )
42		sumfc 14440	. . . . . . 7 |- sum_ x e. A ( ( k e. A |-> B ) ` x ) = sum_ k e. A B
43		fveq2 6191	. . . . . . . 8 |- ( x = ( f ` n ) -> ( ( k e. A |-> B ) ` x ) = ( ( k e. A |-> B ) ` ( f ` n ) ) )
44	24	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ x e. A ) -> ( ( k e. A |-> B ) ` x ) e. CC )
45		f1of 6137	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> f : ( 1 ... ( # ` A ) ) --> A )
46	29, 45	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> f : ( 1 ... ( # ` A ) ) --> A )
47		fvco3 6275	. . . . . . . . 9 |- ( ( f : ( 1 ... ( # ` A ) ) --> A /\ n e. ( 1 ... ( # ` A ) ) ) -> ( ( ( k e. A |-> B ) o. f ) ` n ) = ( ( k e. A |-> B ) ` ( f ` n ) ) )
48	46, 47	sylan 488	. . . . . . . 8 |- ( ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) /\ n e. ( 1 ... ( # ` A ) ) ) -> ( ( ( k e. A |-> B ) o. f ) ` n ) = ( ( k e. A |-> B ) ` ( f ` n ) ) )
49	43, 28, 29, 44, 48	fsum 14451	. . . . . . 7 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> sum_ x e. A ( ( k e. A |-> B ) ` x ) = ( seq 1 ( + , ( ( k e. A |-> B ) o. f ) ) ` ( # ` A ) ) )
50	42, 49	syl5eqr 2670	. . . . . 6 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> sum_ k e. A B = ( seq 1 ( + , ( ( k e. A |-> B ) o. f ) ) ` ( # ` A ) ) )
51	41, 50	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( ( # ` A ) e. NN /\ f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )
52	51	expr 643	. . . 4 |- ( ( ph /\ ( # ` A ) e. NN ) -> ( f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B ) )
53	52	exlimdv 1861	. . 3 |- ( ( ph /\ ( # ` A ) e. NN ) -> ( E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B ) )
54	53	expimpd 629	. 2 |- ( ph -> ( ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B ) )
55		fz1f1o 14441	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
56	19, 55	syl 17	. 2 |- ( ph -> ( A = (/) \/ ( ( # ` A ) e. NN /\ E. f f : ( 1 ... ( # ` A ) ) -1-1-onto-> A ) ) )
57	12, 54, 56	mpjaod 396	1 |- ( ph -> (ℂfld gsumg ( k e. A |-> B ) ) = sum_ k e. A B )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   NNcn 11020   ...cfz 12326   seqcseq 12801   #chash 13117   sum_csu 14416   gsum_g cgsu 16101   Mndcmnd 17294  Cntzccntz 17748  CMndccmn 18193   Ringcrg 18547  ℂ_fldccnfld 19746

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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-supp 7296  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  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  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-cnfld 19747

This theorem is referenced by:  regsumfsum  19814  regsumsupp  19968  plypf1  23968  taylpfval  24119  jensen  24715  amgmlem  24716  lgseisenlem4  25103  esumpfinval  30137  esumpfinvalf  30138  esumpcvgval  30140  esumcvg  30148  sge0tsms  40597  aacllem  42547  amgmwlem  42548  amgmlemALT  42549