Theorem gsum0g 13444

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 2229	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		gsum0.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		eqid 2229	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		id 19	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ 𝑉)
5		0ex 4211	. . . 4 ⊢ ∅ ∈ V
6	5	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅ ∈ V)
7		f0 5518	. . . 4 ⊢ ∅:∅⟶(Base‘𝐺)
8	7	a1i 9	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∅:∅⟶(Base‘𝐺))
9	1, 2, 3, 4, 6, 8	igsumval 13438	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = (℩𝑥((∅ = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))))
10		eqidd 2230	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → ∅ = ∅)
11		eqidd 2230	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 0 = 0 )
12	10, 11	jca 306	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (∅ = ∅ ∧ 0 = 0 ))
13	12	orcd 738	. . 3 ⊢ (𝐺 ∈ 𝑉 → ((∅ = ∅ ∧ 0 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 0 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))))
14		fn0g 13423	. . . . . 6 ⊢ 0g Fn V
15		elex 2811	. . . . . 6 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
16		funfvex 5646	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
17	16	funfni 5423	. . . . . 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 2316	. . . 4 ⊢ (𝐺 ∈ 𝑉 → 0 ∈ V)
20		eueq 2974	. . . . . 6 ⊢ ( 0 ∈ V ↔ ∃!𝑥 𝑥 = 0 )
21		eqid 2229	. . . . . . . . 9 ⊢ ∅ = ∅
22	21	biantrur 303	. . . . . . . 8 ⊢ (𝑥 = 0 ↔ (∅ = ∅ ∧ 𝑥 = 0 ))
23		eluzfz1 10239	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ (𝑚...𝑛))
24		n0i 3497	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑚...𝑛) → ¬ (𝑚...𝑛) = ∅)
25	23, 24	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (𝑚...𝑛) = ∅)
26	25	neqcomd 2234	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ ∅ = (𝑚...𝑛))
27	26	intnanrd 937	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → ¬ (∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛)))
28	27	nrex 2622	. . . . . . . . . 10 ⊢ ¬ ∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
29	28	nex 1546	. . . . . . . . 9 ⊢ ¬ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(∅ = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), ∅)‘𝑛))
30	29	biorfi 751	. . . . . . . 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 2086	. . . . . 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 2236	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0 ))
36	35	anbi2d 464	. . . . . 6 ⊢ (𝑥 = 0 → ((∅ = ∅ ∧ 𝑥 = 0 ) ↔ (∅ = ∅ ∧ 0 = 0 )))
37		eqeq1 2236	. . . . . . . . 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 2531	. . . . . . 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 1871	. . . . . 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 798	. . . . 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 5308	. . . 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 2262	1 ⊢ (𝐺 ∈ 𝑉 → (𝐺 Σg ∅) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395  ∃wex 1538  ∃!weu 2077   ∈ wcel 2200  ∃wrex 2509  Vcvv 2799  ∅c0 3491  ℩cio 5276   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007  ℤ_≥cuz 9733  ...cfz 10216  seqcseq 10681  Basecbs 13047  +_gcplusg 13125  0_gc0g 13304   Σ_g cgsu 13305

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107  ax-pre-ltirr 8122

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-neg 8331  df-inn 9122  df-z 9458  df-uz 9734  df-fz 10217  df-seqfrec 10682  df-ndx 13050  df-slot 13051  df-base 13053  df-0g 13306  df-igsum 13307

This theorem is referenced by:  gsumwsubmcl  13544  gsumwmhm  13546  mulgnn0gsum  13680  gsumfzfsumlem0  14565