Theorem gsum0g 12979

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 2193	. . 3 |- ( Base ` G ) = ( Base ` G )
2		gsum0.z	. . 3 |- .0. = ( 0g ` G )
3		eqid 2193	. . 3 |- ( +g ` G ) = ( +g ` G )
4		id 19	. . 3 |- ( G e. V -> G e. V )
5		0ex 4156	. . . 4 |- (/) e. _V
6	5	a1i 9	. . 3 |- ( G e. V -> (/) e. _V )
7		f0 5444	. . . 4 |- (/) : (/) --> ( Base ` G )
8	7	a1i 9	. . 3 |- ( G e. V -> (/) : (/) --> ( Base ` G ) )
9	1, 2, 3, 4, 6, 8	igsumval 12973	. 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 2194	. . . . 5 |- ( G e. V -> (/) = (/) )
11		eqidd 2194	. . . . 5 |- ( G e. V -> .0. = .0. )
12	10, 11	jca 306	. . . 4 |- ( G e. V -> ( (/) = (/) /\ .0. = .0. ) )
13	12	orcd 734	. . 3 |- ( G e. V -> ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
14		fn0g 12958	. . . . . 6 |- 0g Fn _V
15		elex 2771	. . . . . 6 |- ( G e. V -> G e. _V )
16		funfvex 5571	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
17	16	funfni 5354	. . . . . 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 2280	. . . 4 |- ( G e. V -> .0. e. _V )
20		eueq 2931	. . . . . 6 |- ( .0. e. _V <-> E! x x = .0. )
21		eqid 2193	. . . . . . . . 9 |- (/) = (/)
22	21	biantrur 303	. . . . . . . 8 |- ( x = .0. <-> ( (/) = (/) /\ x = .0. ) )
23		eluzfz1 10097	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m e. ( m ... n ) )
24		n0i 3452	. . . . . . . . . . . . . 14 |- ( m e. ( m ... n ) -> -. ( m ... n ) = (/) )
25	23, 24	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` m ) -> -. ( m ... n ) = (/) )
26	25	neqcomd 2198	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` m ) -> -. (/) = ( m ... n ) )
27	26	intnanrd 933	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` m ) -> -. ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) )
28	27	nrex 2586	. . . . . . . . . 10 |- -. E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
29	28	nex 1511	. . . . . . . . 9 |- -. E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
30	29	biorfi 747	. . . . . . . 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 2051	. . . . . 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 2200	. . . . . . 7 |- ( x = .0. -> ( x = .0. <-> .0. = .0. ) )
36	35	anbi2d 464	. . . . . 6 |- ( x = .0. -> ( ( (/) = (/) /\ x = .0. ) <-> ( (/) = (/) /\ .0. = .0. ) ) )
37		eqeq1 2200	. . . . . . . . 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 2495	. . . . . . 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 1836	. . . . . 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 794	. . . . 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 5244	. . . 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 2226	1 |- ( G e. V -> ( G gsumg (/) ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1503   E!weu 2042   e. wcel 2164   E.wrex 2473   _Vcvv 2760   (/)c0 3446   iotacio 5213   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   Basecbs 12618   +g cplusg 12695   0gc0g 12867   gsum_g cgsu 12868

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-pre-ltirr 7984

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-neg 8193  df-inn 8983  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-igsum 12870

This theorem is referenced by:  gsumwsubmcl  13068  gsumwmhm  13070  mulgnn0gsum  13198  gsumfzfsumlem0  14074