Theorem gsum0g 12982

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 ∈ 𝑉 → (𝐺 Σg ∅) = 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 ∈ 𝑉 → 𝐺 ∈ 𝑉)
5		0ex 4157	. . . 4 ⊢ ∅ ∈ V
6	5	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅ ∈ V)
7		f0 5445	. . . 4 ⊢ ∅:∅⟶(Base‘𝐺)
8	7	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅:∅⟶(Base‘𝐺))
9	1, 2, 3, 4, 6, 8	igsumval 12976	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))))
10		eqidd 2194	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → ∅ = ∅)
11		eqidd 2194	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 0 = 0 )
12	10, 11	jca 306	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (∅ = ∅ ∧ 0 = 0 ))
13	12	orcd 734	. . 3 ⊢ (𝐺 ∈ 𝑉 → ((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
14		fn0g 12961	. . . . . 6 ⊢ 0g Fn V
15		elex 2771	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
16		funfvex 5572	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
17	16	funfni 5355	. . . . . 6 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
18	14, 15, 17	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → (0g‘𝐺) ∈ V)
19	2, 18	eqeltrid 2280	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 0 ∈ V)
20		eueq 2932	. . . . . 6 ⊢ ( 0 ∈ V ↔ ∃!𝑥 𝑥 = 0 )
21		eqid 2193	. . . . . . . . 9 ⊢ ∅ = ∅
22	21	biantrur 303	. . . . . . . 8 ⊢ (𝑥 = 0 ↔ (∅ = ∅ ∧ 𝑥 = 0 ))
23		eluzfz1 10100	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ (𝑚...𝑛))
24		n0i 3453	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑚...𝑛) → ¬ (𝑚...𝑛) = ∅)
25	23, 24	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (𝑚...𝑛) = ∅)
26	25	neqcomd 2198	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ ∅ = (𝑚...𝑛))
27	26	intnanrd 933	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))
28	27	nrex 2586	. . . . . . . . . 10 ⊢ ¬ ∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
29	28	nex 1511	. . . . . . . . 9 ⊢ ¬ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
30	29	biorfi 747	. . . . . . . 8 ⊢ ((∅ = ∅ ∧ 𝑥 = 0 ) ↔ ((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
31	22, 30	bitri 184	. . . . . . 7 ⊢ (𝑥 = 0 ↔ ((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
32	31	eubii 2051	. . . . . 6 ⊢ (∃!𝑥 𝑥 = 0 ↔ ∃!𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
33	20, 32	bitri 184	. . . . 5 ⊢ ( 0 ∈ V ↔ ∃!𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
34	19, 33	sylib 122	. . . 4 ⊢ (𝐺 ∈ 𝑉 → ∃!𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+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 = (seq𝑚((+g‘𝐺), ∅)‘𝑛) ↔ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))
38	37	anbi2d 464	. . . . . . . 8 ⊢ (𝑥 = 0 → ((∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)) ↔ (∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
39	38	rexbidv 2495	. . . . . . 7 ⊢ (𝑥 = 0 → (∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
40	39	exbidv 1836	. . . . . 6 ⊢ (𝑥 = 0 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)) ↔ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
41	36, 40	orbi12d 794	. . . . 5 ⊢ (𝑥 = 0 → (((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))) ↔ ((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))))
42	41	iota2 5245	. . . 4 ⊢ (( 0 ∈ V ∧ ∃!𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))) → (((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))) ↔ (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))) = 0 ))
43	19, 34, 42	syl2anc 411	. . 3 ⊢ (𝐺 ∈ 𝑉 → (((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))) ↔ (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))) = 0 ))
44	13, 43	mpbid 147	. 2 ⊢ (𝐺 ∈ 𝑉 → (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))) = 0 )
45	9, 44	eqtrd 2226	1 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364  ∃wex 1503  ∃!weu 2042   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760  ∅c0 3447  ℩cio 5214   Fn wfn 5250  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  ℤ_≥cuz 9595  ...cfz 10077  seqcseq 10521  Basecbs 12621  +_gcplusg 12698  0_gc0g 12870   Σ_g cgsu 12871

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 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971  ax-pre-ltirr 7986

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-neg 8195  df-inn 8985  df-z 9321  df-uz 9596  df-fz 10078  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-0g 12872  df-igsum 12873

This theorem is referenced by:  gsumwsubmcl  13071  gsumwmhm  13073  mulgnn0gsum  13201  gsumfzfsumlem0  14085