Theorem gsum0g 13478

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 2231	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		gsum0.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		eqid 2231	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		id 19	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ 𝑉)
5		0ex 4216	. . . 4 ⊢ ∅ ∈ V
6	5	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅ ∈ V)
7		f0 5527	. . . 4 ⊢ ∅:∅⟶(Base‘𝐺)
8	7	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅:∅⟶(Base‘𝐺))
9	1, 2, 3, 4, 6, 8	igsumval 13472	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))))
10		eqidd 2232	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → ∅ = ∅)
11		eqidd 2232	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 0 = 0 )
12	10, 11	jca 306	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (∅ = ∅ ∧ 0 = 0 ))
13	12	orcd 740	. . 3 ⊢ (𝐺 ∈ 𝑉 → ((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
14		fn0g 13457	. . . . . 6 ⊢ 0g Fn V
15		elex 2814	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
16		funfvex 5656	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
17	16	funfni 5432	. . . . . 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 2318	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 0 ∈ V)
20		eueq 2977	. . . . . 6 ⊢ ( 0 ∈ V ↔ ∃!𝑥 𝑥 = 0 )
21		eqid 2231	. . . . . . . . 9 ⊢ ∅ = ∅
22	21	biantrur 303	. . . . . . . 8 ⊢ (𝑥 = 0 ↔ (∅ = ∅ ∧ 𝑥 = 0 ))
23		eluzfz1 10265	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ (𝑚...𝑛))
24		n0i 3500	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑚...𝑛) → ¬ (𝑚...𝑛) = ∅)
25	23, 24	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (𝑚...𝑛) = ∅)
26	25	neqcomd 2236	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ ∅ = (𝑚...𝑛))
27	26	intnanrd 939	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))
28	27	nrex 2624	. . . . . . . . . 10 ⊢ ¬ ∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
29	28	nex 1548	. . . . . . . . 9 ⊢ ¬ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
30	29	biorfi 753	. . . . . . . 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 2088	. . . . . 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 2238	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0 ))
36	35	anbi2d 464	. . . . . 6 ⊢ (𝑥 = 0 → ((∅ = ∅ ∧ 𝑥 = 0 ) ↔ (∅ = ∅ ∧ 0 = 0 )))
37		eqeq1 2238	. . . . . . . . 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 2533	. . . . . . 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 1873	. . . . . 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 800	. . . . 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 5316	. . . 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 2264	1 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397  ∃wex 1540  ∃!weu 2079   ∈ wcel 2202  ∃wrex 2511  Vcvv 2802  ∅c0 3494  ℩cio 5284   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  ℤ_≥cuz 9754  ...cfz 10242  seqcseq 10708  Basecbs 13081  +_gcplusg 13159  0_gc0g 13338   Σ_g cgsu 13339

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-pre-ltirr 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-neg 8352  df-inn 9143  df-z 9479  df-uz 9755  df-fz 10243  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-0g 13340  df-igsum 13341

This theorem is referenced by:  gsumwsubmcl  13578  gsumwmhm  13580  mulgnn0gsum  13714  gsumfzfsumlem0  14599  gfsumval  16680  gfsum0  16682  gsumgfsum  16684