Theorem gsum0g 13609

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 2232	. . 3 |- ( Base ` G ) = ( Base ` G )
2		gsum0.z	. . 3 |- .0. = ( 0g ` G )
3		eqid 2232	. . 3 |- ( +g ` G ) = ( +g ` G )
4		id 19	. . 3 |- ( G e. V -> G e. V )
5		0ex 4237	. . . 4 |- (/) e. _V
6	5	a1i 9	. . 3 |- ( G e. V -> (/) e. _V )
7		f0 5558	. . . 4 |- (/) : (/) --> ( Base ` G )
8	7	a1i 9	. . 3 |- ( G e. V -> (/) : (/) --> ( Base ` G ) )
9	1, 2, 3, 4, 6, 8	igsumval 13603	. 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 2233	. . . . 5 |- ( G e. V -> (/) = (/) )
11		eqidd 2233	. . . . 5 |- ( G e. V -> .0. = .0. )
12	10, 11	jca 306	. . . 4 |- ( G e. V -> ( (/) = (/) /\ .0. = .0. ) )
13	12	orcd 741	. . 3 |- ( G e. V -> ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
14		fn0g 13588	. . . . . 6 |- 0g Fn _V
15		elex 2825	. . . . . 6 |- ( G e. V -> G e. _V )
16		funfvex 5687	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
17	16	funfni 5458	. . . . . 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 2319	. . . 4 |- ( G e. V -> .0. e. _V )
20		eueq 2988	. . . . . 6 |- ( .0. e. _V <-> E! x x = .0. )
21		eqid 2232	. . . . . . . . 9 |- (/) = (/)
22	21	biantrur 303	. . . . . . . 8 |- ( x = .0. <-> ( (/) = (/) /\ x = .0. ) )
23		eluzfz1 10365	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m e. ( m ... n ) )
24		n0i 3514	. . . . . . . . . . . . . 14 |- ( m e. ( m ... n ) -> -. ( m ... n ) = (/) )
25	23, 24	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` m ) -> -. ( m ... n ) = (/) )
26	25	neqcomd 2237	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` m ) -> -. (/) = ( m ... n ) )
27	26	intnanrd 940	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` m ) -> -. ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) )
28	27	nrex 2634	. . . . . . . . . 10 |- -. E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
29	28	nex 1549	. . . . . . . . 9 |- -. E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
30	29	biorfi 754	. . . . . . . 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 2089	. . . . . 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 2239	. . . . . . 7 |- ( x = .0. -> ( x = .0. <-> .0. = .0. ) )
36	35	anbi2d 464	. . . . . 6 |- ( x = .0. -> ( ( (/) = (/) /\ x = .0. ) <-> ( (/) = (/) /\ .0. = .0. ) ) )
37		eqeq1 2239	. . . . . . . . 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 2543	. . . . . . 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 1874	. . . . . 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 801	. . . . 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 5342	. . . 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 2265	1 |- ( G e. V -> ( G gsumg (/) ) = .0. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   E!weu 2080   e. wcel 2203   E.wrex 2521   _Vcvv 2813   (/)c0 3508   iotacio 5310   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   ZZ>=cuz 9853   ...cfz 10342   seqcseq 10809   Basecbs 13212   +g cplusg 13290   0gc0g 13469   gsum_g cgsu 13470

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-pre-ltirr 8239

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-neg 8447  df-inn 9238  df-z 9578  df-uz 9854  df-fz 10343  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471  df-igsum 13472

This theorem is referenced by:  gsumwsubmcl  13709  gsumwmhm  13711  mulgnn0gsum  13845  gsumsplit0  14063  gsumfzfsumlem0  14734  gfsumval  16862  gfsum0  16864  gsumgfsum  16866