Theorem gsum0g 13469

Description: Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypothesis

Ref	Expression
gsum0.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
gsum0g	|- ( G e. V -> ( G gsumg (/) ) = .0. )

Proof of Theorem gsum0g

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

Step	Hyp	Ref	Expression
1		eqid 2229	. . 3 |- ( Base ` G ) = ( Base ` G )
2		gsum0.z	. . 3 |- .0. = ( 0g ` G )
3		eqid 2229	. . 3 |- ( +g ` G ) = ( +g ` G )
4		id 19	. . 3 |- ( G e. V -> G e. V )
5		0ex 4214	. . . 4 |- (/) e. _V
6	5	a1i 9	. . 3 |- ( G e. V -> (/) e. _V )
7		f0 5524	. . . 4 |- (/) : (/) --> ( Base ` G )
8	7	a1i 9	. . 3 |- ( G e. V -> (/) : (/) --> ( Base ` G ) )
9	1, 2, 3, 4, 6, 8	igsumval 13463	. 2 |- ( G e. V -> ( G gsumg (/) ) = ( iota x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) )
10		eqidd 2230	. . . . 5 |- ( G e. V -> (/) = (/) )
11		eqidd 2230	. . . . 5 |- ( G e. V -> .0. = .0. )
12	10, 11	jca 306	. . . 4 |- ( G e. V -> ( (/) = (/) /\ .0. = .0. ) )
13	12	orcd 738	. . 3 |- ( G e. V -> ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
14		fn0g 13448	. . . . . 6 |- 0g Fn _V
15		elex 2812	. . . . . 6 |- ( G e. V -> G e. _V )
16		funfvex 5652	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
17	16	funfni 5429	. . . . . 6 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
18	14, 15, 17	sylancr 414	. . . . 5 |- ( G e. V -> ( 0g ` G ) e. _V )
19	2, 18	eqeltrid 2316	. . . 4 |- ( G e. V -> .0. e. _V )
20		eueq 2975	. . . . . 6 |- ( .0. e. _V <-> E! x x = .0. )
21		eqid 2229	. . . . . . . . 9 |- (/) = (/)
22	21	biantrur 303	. . . . . . . 8 |- ( x = .0. <-> ( (/) = (/) /\ x = .0. ) )
23		eluzfz1 10256	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m e. ( m ... n ) )
24		n0i 3498	. . . . . . . . . . . . . 14 |- ( m e. ( m ... n ) -> -. ( m ... n ) = (/) )
25	23, 24	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` m ) -> -. ( m ... n ) = (/) )
26	25	neqcomd 2234	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` m ) -> -. (/) = ( m ... n ) )
27	26	intnanrd 937	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` m ) -> -. ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) )
28	27	nrex 2622	. . . . . . . . . 10 |- -. E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
29	28	nex 1546	. . . . . . . . 9 |- -. E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
30	29	biorfi 751	. . . . . . . 8 |- ( ( (/) = (/) /\ x = .0. ) <-> ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
31	22, 30	bitri 184	. . . . . . 7 |- ( x = .0. <-> ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
32	31	eubii 2086	. . . . . 6 |- ( E! x x = .0. <-> E! x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
33	20, 32	bitri 184	. . . . 5 |- ( .0. e. _V <-> E! x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
34	19, 33	sylib 122	. . . 4 |- ( G e. V -> E! x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
35		eqeq1 2236	. . . . . . 7 |- ( x = .0. -> ( x = .0. <-> .0. = .0. ) )
36	35	anbi2d 464	. . . . . 6 |- ( x = .0. -> ( ( (/) = (/) /\ x = .0. ) <-> ( (/) = (/) /\ .0. = .0. ) ) )
37		eqeq1 2236	. . . . . . . . 9 |- ( x = .0. -> ( x = ( seq m ( ( +g ` G ) , (/) ) ` n ) <-> .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) )
38	37	anbi2d 464	. . . . . . . 8 |- ( x = .0. -> ( ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) <-> ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
39	38	rexbidv 2531	. . . . . . 7 |- ( x = .0. -> ( E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) <-> E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
40	39	exbidv 1871	. . . . . 6 |- ( x = .0. -> ( E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) <-> E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
41	36, 40	orbi12d 798	. . . . 5 |- ( x = .0. -> ( ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) <-> ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) )
42	41	iota2 5314	. . . 4 |- ( ( .0. e. _V /\ E! x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) -> ( ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) <-> ( iota x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) = .0. ) )
43	19, 34, 42	syl2anc 411	. . 3 |- ( G e. V -> ( ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) <-> ( iota x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) = .0. ) )
44	13, 43	mpbid 147	. 2 |- ( G e. V -> ( iota x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) = .0. )
45	9, 44	eqtrd 2262	1 |- ( G e. V -> ( G gsumg (/) ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   E.wex 1538   E!weu 2077   e. wcel 2200   E.wrex 2509   _Vcvv 2800   (/)c0 3492   iotacio 5282   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   ZZ>=cuz 9745   ...cfz 10233   seqcseq 10699   Basecbs 13072   +g cplusg 13150   0gc0g 13329   gsum_g cgsu 13330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119  ax-pre-ltirr 8134

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-neg 8343  df-inn 9134  df-z 9470  df-uz 9746  df-fz 10234  df-seqfrec 10700  df-ndx 13075  df-slot 13076  df-base 13078  df-0g 13331  df-igsum 13332

This theorem is referenced by:  gsumwsubmcl  13569  gsumwmhm  13571  mulgnn0gsum  13705  gsumfzfsumlem0  14590