Theorem gsum0g 13659

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 2234	. . 3 |- ( Base ` G ) = ( Base ` G )
2		gsum0.z	. . 3 |- .0. = ( 0g ` G )
3		eqid 2234	. . 3 |- ( +g ` G ) = ( +g ` G )
4		id 19	. . 3 |- ( G e. V -> G e. V )
5		0ex 4242	. . . 4 |- (/) e. _V
6	5	a1i 9	. . 3 |- ( G e. V -> (/) e. _V )
7		f0 5563	. . . 4 |- (/) : (/) --> ( Base ` G )
8	7	a1i 9	. . 3 |- ( G e. V -> (/) : (/) --> ( Base ` G ) )
9	1, 2, 3, 4, 6, 8	igsumval 13653	. 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 2235	. . . . 5 |- ( G e. V -> (/) = (/) )
11		eqidd 2235	. . . . 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 13638	. . . . . 6 |- 0g Fn _V
15		elex 2827	. . . . . 6 |- ( G e. V -> G e. _V )
16		funfvex 5692	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
17	16	funfni 5463	. . . . . 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 2321	. . . 4 |- ( G e. V -> .0. e. _V )
20		eueq 2991	. . . . . 6 |- ( .0. e. _V <-> E! x x = .0. )
21		eqid 2234	. . . . . . . . 9 |- (/) = (/)
22	21	biantrur 303	. . . . . . . 8 |- ( x = .0. <-> ( (/) = (/) /\ x = .0. ) )
23		eluzfz1 10385	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m e. ( m ... n ) )
24		n0i 3518	. . . . . . . . . . . . . 14 |- ( m e. ( m ... n ) -> -. ( m ... n ) = (/) )
25	23, 24	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` m ) -> -. ( m ... n ) = (/) )
26	25	neqcomd 2239	. . . . . . . . . . . 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 2636	. . . . . . . . . 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 2091	. . . . . 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 2241	. . . . . . 7 |- ( x = .0. -> ( x = .0. <-> .0. = .0. ) )
36	35	anbi2d 464	. . . . . 6 |- ( x = .0. -> ( ( (/) = (/) /\ x = .0. ) <-> ( (/) = (/) /\ .0. = .0. ) ) )
37		eqeq1 2241	. . . . . . . . 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 2545	. . . . . . 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 5347	. . . 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 2267	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 2082   e. wcel 2205   E.wrex 2523   _Vcvv 2815   (/)c0 3512   iotacio 5315   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240  ax-pre-ltirr 8255

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-neg 8463  df-inn 9255  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsumwsubmcl  13751  gsumwmhm  13753  mulgnn0gsum  13881  gsumsplit0  14099  gfsumval  14102  gfsum0  14104  gsumgfsum  14106  gsumfzfsumlem0  14860