Theorem gsum0g 13303

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 2206	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		gsum0.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		eqid 2206	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		id 19	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ 𝑉)
5		0ex 4179	. . . 4 ⊢ ∅ ∈ V
6	5	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅ ∈ V)
7		f0 5478	. . . 4 ⊢ ∅:∅⟶(Base‘𝐺)
8	7	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅:∅⟶(Base‘𝐺))
9	1, 2, 3, 4, 6, 8	igsumval 13297	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))))
10		eqidd 2207	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → ∅ = ∅)
11		eqidd 2207	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 0 = 0 )
12	10, 11	jca 306	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (∅ = ∅ ∧ 0 = 0 ))
13	12	orcd 735	. . 3 ⊢ (𝐺 ∈ 𝑉 → ((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
14		fn0g 13282	. . . . . 6 ⊢ 0g Fn V
15		elex 2785	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
16		funfvex 5606	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
17	16	funfni 5385	. . . . . 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 2293	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 0 ∈ V)
20		eueq 2948	. . . . . 6 ⊢ ( 0 ∈ V ↔ ∃!𝑥 𝑥 = 0 )
21		eqid 2206	. . . . . . . . 9 ⊢ ∅ = ∅
22	21	biantrur 303	. . . . . . . 8 ⊢ (𝑥 = 0 ↔ (∅ = ∅ ∧ 𝑥 = 0 ))
23		eluzfz1 10173	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ (𝑚...𝑛))
24		n0i 3470	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑚...𝑛) → ¬ (𝑚...𝑛) = ∅)
25	23, 24	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (𝑚...𝑛) = ∅)
26	25	neqcomd 2211	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ ∅ = (𝑚...𝑛))
27	26	intnanrd 934	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))
28	27	nrex 2599	. . . . . . . . . 10 ⊢ ¬ ∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
29	28	nex 1524	. . . . . . . . 9 ⊢ ¬ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
30	29	biorfi 748	. . . . . . . 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 2064	. . . . . 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 2213	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0 ))
36	35	anbi2d 464	. . . . . 6 ⊢ (𝑥 = 0 → ((∅ = ∅ ∧ 𝑥 = 0 ) ↔ (∅ = ∅ ∧ 0 = 0 )))
37		eqeq1 2213	. . . . . . . . 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 2508	. . . . . . 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 1849	. . . . . 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 795	. . . . 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 5270	. . . 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 2239	1 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773  ∅c0 3464  ℩cio 5239   Fn wfn 5275  ⟶wf 5276  ‘cfv 5280  (class class class)co 5957  ℤ_≥cuz 9668  ...cfz 10150  seqcseq 10614  Basecbs 12907  +_gcplusg 12984  0_gc0g 13163   Σ_g cgsu 13164

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042  ax-pre-ltirr 8057

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-recs 6404  df-frec 6490  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-neg 8266  df-inn 9057  df-z 9393  df-uz 9669  df-fz 10151  df-seqfrec 10615  df-ndx 12910  df-slot 12911  df-base 12913  df-0g 13165  df-igsum 13166

This theorem is referenced by:  gsumwsubmcl  13403  gsumwmhm  13405  mulgnn0gsum  13539  gsumfzfsumlem0  14423