Theorem igsumvalx 12972

Description: Expand out the substitutions in df-igsum 12870. (Contributed by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
gsumval.b	⊢ 𝐵 = (Base‘𝐺)
gsumval.z	⊢ 0 = (0g‘𝐺)
gsumval.p	⊢ + = (+g‘𝐺)
gsumval.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumvalx.f	⊢ (𝜑 → 𝐹 ∈ 𝑋)
gsumvalx.a	⊢ (𝜑 → dom 𝐹 = 𝐴)

Assertion

Ref	Expression
igsumvalx	⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))

Distinct variable groups:   𝑥, +   𝑥, 0   𝑚,𝐹,𝑛,𝑥   𝑚,𝐺,𝑛,𝑥   𝜑,𝑚,𝑛,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑚,𝑛)   𝐵(𝑥,𝑚,𝑛)   + (𝑚,𝑛)   𝑉(𝑥,𝑚,𝑛)   𝑋(𝑥,𝑚,𝑛)   0 (𝑚,𝑛)

Proof of Theorem igsumvalx

Dummy variables 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-igsum 12870	. . 3 ⊢ Σg = (𝑤 ∈ V, 𝑔 ∈ V ↦ (℩𝑥((dom 𝑔 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛)))))
2	1	a1i 9	. 2 ⊢ (𝜑 → Σg = (𝑤 ∈ V, 𝑔 ∈ V ↦ (℩𝑥((dom 𝑔 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛))))))
3		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑔 = 𝐹)
4	3	dmeqd 4864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → dom 𝑔 = dom 𝐹)
5		gsumvalx.a	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → dom 𝐹 = 𝐴)
7	4, 6	eqtrd 2226	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → dom 𝑔 = 𝐴)
8	7	eqeq1d 2202	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (dom 𝑔 = ∅ ↔ 𝐴 = ∅))
9		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑤 = 𝐺)
10	9	fveq2d 5558	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (0g‘𝑤) = (0g‘𝐺))
11		gsumval.z	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	10, 11	eqtr4di 2244	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (0g‘𝑤) = 0 )
13	12	eqeq2d 2205	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (𝑥 = (0g‘𝑤) ↔ 𝑥 = 0 ))
14	8, 13	anbi12d 473	. . . 4 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → ((dom 𝑔 = ∅ ∧ 𝑥 = (0g‘𝑤)) ↔ (𝐴 = ∅ ∧ 𝑥 = 0 )))
15	7	eqeq1d 2202	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (dom 𝑔 = (𝑚...𝑛) ↔ 𝐴 = (𝑚...𝑛)))
16		eqidd 2194	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑚 = 𝑚)
17	9	fveq2d 5558	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (+g‘𝑤) = (+g‘𝐺))
18		gsumval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
19	17, 18	eqtr4di 2244	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (+g‘𝑤) = + )
20	16, 19, 3	seqeq123d 10527	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → seq𝑚((+g‘𝑤), 𝑔) = seq𝑚( + , 𝐹))
21	20	fveq1d 5556	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (seq𝑚((+g‘𝑤), 𝑔)‘𝑛) = (seq𝑚( + , 𝐹)‘𝑛))
22	21	eqeq2d 2205	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛) ↔ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
23	15, 22	anbi12d 473	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → ((dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
24	23	rexbidv 2495	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
25	24	exbidv 1836	. . . 4 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛)) ↔ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
26	14, 25	orbi12d 794	. . 3 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (((dom 𝑔 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛))) ↔ ((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
27	26	iotabidv 5237	. 2 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (℩𝑥((dom 𝑔 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑔 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛)))) = (℩𝑥((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
28		gsumval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
29	28	elexd 2773	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
30		gsumvalx.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
31	30	elexd 2773	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
32		unab 3426	. . . 4 ⊢ ({𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∪ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))}) = {𝑥 ∣ ((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))}
33		df-sn 3624	. . . . . . 7 ⊢ { 0 } = {𝑥 ∣ 𝑥 = 0 }
34		fn0g 12958	. . . . . . . . . 10 ⊢ 0g Fn V
35		funfvex 5571	. . . . . . . . . . 11 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
36	35	funfni 5354	. . . . . . . . . 10 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
37	34, 29, 36	sylancr 414	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
38	11, 37	eqeltrid 2280	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
39		snexg 4213	. . . . . . . 8 ⊢ ( 0 ∈ V → { 0 } ∈ V)
40	38, 39	syl 14	. . . . . . 7 ⊢ (𝜑 → { 0 } ∈ V)
41	33, 40	eqeltrrid 2281	. . . . . 6 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 0 } ∈ V)
42		simpr 110	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∧ 𝑥 = 0 ) → 𝑥 = 0 )
43	42	ss2abi 3251	. . . . . . 7 ⊢ {𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ⊆ {𝑥 ∣ 𝑥 = 0 }
44	43	a1i 9	. . . . . 6 ⊢ (𝜑 → {𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ⊆ {𝑥 ∣ 𝑥 = 0 })
45	41, 44	ssexd 4169	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∈ V)
46		zex 9326	. . . . . . 7 ⊢ ℤ ∈ V
47	46, 46	ab2rexex 6183	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)} ∈ V
48		df-rex 2478	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛(𝑛 ∈ (ℤ≥‘𝑚) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
49		eluzel2 9597	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
50		eluzelz 9601	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑛 ∈ ℤ)
51	49, 50	jca 306	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ))
52		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
53	51, 52	anim12i 338	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
54		anass 401	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
55	53, 54	sylib 122	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
56	55	eximi 1611	. . . . . . . . . . . 12 ⊢ (∃𝑛(𝑛 ∈ (ℤ≥‘𝑚) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ∃𝑛(𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
57	48, 56	sylbi 121	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → ∃𝑛(𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
58		19.42v 1918	. . . . . . . . . . 11 ⊢ (∃𝑛(𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) ↔ (𝑚 ∈ ℤ ∧ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
59	57, 58	sylib 122	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (𝑚 ∈ ℤ ∧ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
60		df-rex 2478	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
61	60	anbi2i 457	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝑚 ∈ ℤ ∧ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
62	59, 61	sylibr 134	. . . . . . . . 9 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → (𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
63	62	eximi 1611	. . . . . . . 8 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
64		df-rex 2478	. . . . . . . 8 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
65	63, 64	sylibr 134	. . . . . . 7 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
66	65	ss2abi 3251	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))} ⊆ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)}
67	47, 66	ssexi 4167	. . . . 5 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))} ∈ V
68		unexg 4474	. . . . 5 ⊢ (({𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∈ V ∧ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))} ∈ V) → ({𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∪ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))}) ∈ V)
69	45, 67, 68	sylancl 413	. . . 4 ⊢ (𝜑 → ({𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∪ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))}) ∈ V)
70	32, 69	eqeltrrid 2281	. . 3 ⊢ (𝜑 → {𝑥 ∣ ((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))} ∈ V)
71		iotaexab 5233	. . 3 ⊢ ({𝑥 ∣ ((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))} ∈ V → (℩𝑥((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))) ∈ V)
72	70, 71	syl 14	. 2 ⊢ (𝜑 → (℩𝑥((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))) ∈ V)
73	2, 27, 29, 31, 72	ovmpod 6046	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∃wrex 2473  Vcvv 2760   ∪ cun 3151   ⊆ wss 3153  ∅c0 3446  {csn 3618  dom cdm 4659  ℩cio 5213   Fn wfn 5249  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  ℤcz 9317  ℤ_≥cuz 9592  ...cfz 10074  seqcseq 10518  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867   Σ_g cgsu 12868

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-recs 6358  df-frec 6444  df-neg 8193  df-inn 8983  df-z 9318  df-uz 9593  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-igsum 12870

This theorem is referenced by:  igsumval  12973  gsumpropd  12975  gsumpropd2  12976