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 ∈ 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 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 ∈ 𝑉 → 𝐺 ∈ 𝑉)
5		0ex 4242	. . . 4 ⊢ ∅ ∈ V
6	5	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅ ∈ V)
7		f0 5563	. . . 4 ⊢ ∅:∅⟶(Base‘𝐺)
8	7	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅:∅⟶(Base‘𝐺))
9	1, 2, 3, 4, 6, 8	igsumval 13653	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))))
10		eqidd 2235	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → ∅ = ∅)
11		eqidd 2235	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 0 = 0 )
12	10, 11	jca 306	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (∅ = ∅ ∧ 0 = 0 ))
13	12	orcd 741	. . 3 ⊢ (𝐺 ∈ 𝑉 → ((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
14		fn0g 13638	. . . . . 6 ⊢ 0g Fn V
15		elex 2827	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
16		funfvex 5692	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
17	16	funfni 5463	. . . . . 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 2321	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 0 ∈ V)
20		eueq 2991	. . . . . 6 ⊢ ( 0 ∈ V ↔ ∃!𝑥 𝑥 = 0 )
21		eqid 2234	. . . . . . . . 9 ⊢ ∅ = ∅
22	21	biantrur 303	. . . . . . . 8 ⊢ (𝑥 = 0 ↔ (∅ = ∅ ∧ 𝑥 = 0 ))
23		eluzfz1 10385	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ (𝑚...𝑛))
24		n0i 3518	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑚...𝑛) → ¬ (𝑚...𝑛) = ∅)
25	23, 24	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (𝑚...𝑛) = ∅)
26	25	neqcomd 2239	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ ∅ = (𝑚...𝑛))
27	26	intnanrd 940	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))
28	27	nrex 2636	. . . . . . . . . 10 ⊢ ¬ ∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
29	28	nex 1549	. . . . . . . . 9 ⊢ ¬ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
30	29	biorfi 754	. . . . . . . 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 2091	. . . . . 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 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 = (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 2545	. . . . . . 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 1874	. . . . . 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 801	. . . . 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 5347	. . . 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 2267	1 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541  ∃!weu 2082   ∈ wcel 2205  ∃wrex 2523  Vcvv 2815  ∅c0 3512  ℩cio 5315   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  ℤ_≥cuz 9871  ...cfz 10361  seqcseq 10833  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553   Σ_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