Theorem gzsum0 13772

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
gzsum0	|- ( G e. V -> ( G gzsumgz (/) ) = .0. )

Proof of Theorem gzsum0

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

Step	Hyp	Ref	Expression
1		eqid 2238	. . 3 |- ( Base ` G ) = ( Base ` G )
2		gsum0.z	. . 3 |- .0. = ( 0g ` G )
3		eqid 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
4		id 19	. . 3 |- ( G e. V -> G e. V )
5		0ex 4262	. . . 4 |- (/) e. _V
6	5	a1i 9	. . 3 |- ( G e. V -> (/) e. _V )
7		f0 5586	. . . 4 |- (/) : (/) --> ( Base ` G )
8	7	a1i 9	. . 3 |- ( G e. V -> (/) : (/) --> ( Base ` G ) )
9	1, 2, 3, 4, 6, 8	gzsumval 13769	. 2 |- ( G e. V -> ( G gzsumgz (/) ) = ( iota x ( ( (/) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) ) )
10		eqidd 2239	. . . . 5 |- ( G e. V -> (/) = (/) )
11		eqidd 2239	. . . . 5 |- ( G e. V -> .0. = .0. )
12	10, 11	jca 306	. . . 4 |- ( G e. V -> ( (/) = (/) /\ .0. = .0. ) )
13	12	orcd 745	. . 3 |- ( G e. V -> ( ( (/) = (/) /\ .0. = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
14		fn0g 13754	. . . . . 6 |- 0g Fn _V
15		elex 2833	. . . . . 6 |- ( G e. V -> G e. _V )
16		funfvex 5715	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
17	16	funfni 5486	. . . . . 6 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
18	14, 15, 17	sylancr 418	. . . . 5 |- ( G e. V -> ( 0g ` G ) e. _V )
19	2, 18	eqeltrid 2325	. . . 4 |- ( G e. V -> .0. e. _V )
20		eueq 2997	. . . . . 6 |- ( .0. e. _V <-> E! x x = .0. )
21		eqid 2238	. . . . . . . . 9 |- (/) = (/)
22	21	biantrur 303	. . . . . . . 8 |- ( x = .0. <-> ( (/) = (/) /\ x = .0. ) )
23		eluzfz1 10452	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m e. ( m ... n ) )
24		n0i 3527	. . . . . . . . . . . . . 14 |- ( m e. ( m ... n ) -> -. ( m ... n ) = (/) )
25	23, 24	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( ZZ>= ` m ) -> -. ( m ... n ) = (/) )
26	25	neqcomd 2243	. . . . . . . . . . . 12 |- ( n e. ( ZZ>= ` m ) -> -. (/) = ( m ... n ) )
27	26	intnanrd 944	. . . . . . . . . . 11 |- ( n e. ( ZZ>= ` m ) -> -. ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) )
28	27	nrex 2642	. . . . . . . . . 10 |- -. E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
29	28	nex 1553	. . . . . . . . 9 |- -. E. m E. n e. ( ZZ>= ` m ) ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) )
30	29	biorfi 758	. . . . . . . 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 2095	. . . . . 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 2245	. . . . . . 7 |- ( x = .0. -> ( x = .0. <-> .0. = .0. ) )
36	35	anbi2d 468	. . . . . 6 |- ( x = .0. -> ( ( (/) = (/) /\ x = .0. ) <-> ( (/) = (/) /\ .0. = .0. ) ) )
37		eqeq1 2245	. . . . . . . . 9 |- ( x = .0. -> ( x = ( seq m ( ( +g ` G ) , (/) ) ` n ) <-> .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) )
38	37	anbi2d 468	. . . . . . . 8 |- ( x = .0. -> ( ( (/) = ( m ... n ) /\ x = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) <-> ( (/) = ( m ... n ) /\ .0. = ( seq m ( ( +g ` G ) , (/) ) ` n ) ) ) )
39	38	rexbidv 2551	. . . . . . 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 1878	. . . . . 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 805	. . . . 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 5370	. . . 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 415	. . 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 2271	1 |- ( G e. V -> ( G gzsumgz (/) ) = .0. )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   E.wrex 2529   _Vcvv 2821   (/)c0 3520   iotacio 5338   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   ZZ>=cuz 9937   ...cfz 10428   seqcseq 10905   Basecbs 13410   +g cplusg 13490   0gc0g 13669   gzsum_gz cgzsu 13670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282  ax-pre-ltirr 8297

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-neg 8507  df-inn 9314  df-z 9656  df-uz 9938  df-fz 10429  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-0g 13671  df-gzsum 13672

This theorem is used by:  gzsumwsubmcl  13860  gzsumwmhm  13862  mulgnn0gzsum  13990  gzsumsplit0  14238  gsumvalfi  14242  gsum0cmn  14244  gzsumgsum  14245