Theorem gzsumvalx 13686

Description: Expand out the substitutions in df-gzsum 13590. (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
gzsumvalx	⊢ (𝜑 → (𝐺 Σgz 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 gzsumvalx

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

Step	Hyp	Ref	Expression
1		df-gzsum 13590	. . 3 ⊢ Σgz = (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 → Σgz = (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 537	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑔 = 𝐹)
4	3	dmeqd 4978	. . . . . . 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 2271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → dom 𝑔 = 𝐴)
8	7	eqeq1d 2247	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (dom 𝑔 = ∅ ↔ 𝐴 = ∅))
9		simprl 535	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑤 = 𝐺)
10	9	fveq2d 5694	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (0g‘𝑤) = (0g‘𝐺))
11		gsumval.z	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
12	10, 11	eqtr4di 2289	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (0g‘𝑤) = 0 )
13	12	eqeq2d 2250	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (𝑥 = (0g‘𝑤) ↔ 𝑥 = 0 ))
14	8, 13	anbi12d 477	. . . 4 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → ((dom 𝑔 = ∅ ∧ 𝑥 = (0g‘𝑤)) ↔ (𝐴 = ∅ ∧ 𝑥 = 0 )))
15	7	eqeq1d 2247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (dom 𝑔 = (𝑚...𝑛) ↔ 𝐴 = (𝑚...𝑛)))
16		eqidd 2239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → 𝑚 = 𝑚)
17	9	fveq2d 5694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (+g‘𝑤) = (+g‘𝐺))
18		gsumval.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
19	17, 18	eqtr4di 2289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (+g‘𝑤) = + )
20	16, 19, 3	seqeq123d 10871	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → seq𝑚((+g‘𝑤), 𝑔) = seq𝑚( + , 𝐹))
21	20	fveq1d 5692	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (seq𝑚((+g‘𝑤), 𝑔)‘𝑛) = (seq𝑚( + , 𝐹)‘𝑛))
22	21	eqeq2d 2250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 = 𝐺 ∧ 𝑔 = 𝐹)) → (𝑥 = (seq𝑚((+g‘𝑤), 𝑔)‘𝑛) ↔ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
23	15, 22	anbi12d 477	. . . . . 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 2551	. . . . 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 1878	. . . 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 805	. . 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 5355	. 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 2835	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
30		gsumvalx.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑋)
31	30	elexd 2835	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
32		unab 3498	. . . 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 3711	. . . . . . 7 ⊢ { 0 } = {𝑥 ∣ 𝑥 = 0 }
34		fn0g 13672	. . . . . . . . . 10 ⊢ 0g Fn V
35		funfvex 5707	. . . . . . . . . . 11 ⊢ ((Fun 0g ∧ 𝐺 ∈ dom 0g) → (0g‘𝐺) ∈ V)
36	35	funfni 5478	. . . . . . . . . 10 ⊢ ((0g Fn V ∧ 𝐺 ∈ V) → (0g‘𝐺) ∈ V)
37	34, 29, 36	sylancr 418	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐺) ∈ V)
38	11, 37	eqeltrid 2325	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ V)
39		snexg 4316	. . . . . . . 8 ⊢ ( 0 ∈ V → { 0 } ∈ V)
40	38, 39	syl 14	. . . . . . 7 ⊢ (𝜑 → { 0 } ∈ V)
41	33, 40	eqeltrrid 2326	. . . . . 6 ⊢ (𝜑 → {𝑥 ∣ 𝑥 = 0 } ∈ V)
42		simpr 110	. . . . . . . 8 ⊢ ((𝐴 = ∅ ∧ 𝑥 = 0 ) → 𝑥 = 0 )
43	42	ss2abi 3320	. . . . . . 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 4268	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∈ V)
46		zex 9632	. . . . . . 7 ⊢ ℤ ∈ V
47	46, 46	ab2rexex 6354	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)} ∈ V
48		df-rex 2534	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛(𝑛 ∈ (ℤ≥‘𝑚) ∧ (𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
49		eluzel2 9905	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
50		eluzelz 9910	. . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . 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 1653	. . . . . . . . . . . 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 1962	. . . . . . . . . . 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 2534	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
61	60	anbi2i 461	. . . . . . . . . 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 1653	. . . . . . . 8 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
64		df-rex 2534	. . . . . . . 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 3320	. . . . . 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 4266	. . . . 5 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))} ∈ V
68		unexg 4584	. . . . 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 417	. . . 4 ⊢ (𝜑 → ({𝑥 ∣ (𝐴 = ∅ ∧ 𝑥 = 0 )} ∪ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))}) ∈ V)
70	32, 69	eqeltrrid 2326	. . 3 ⊢ (𝜑 → {𝑥 ∣ ((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))} ∈ V)
71		iotaexab 5351	. . 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 6206	1 ⊢ (𝜑 → (𝐺 Σgz 𝐹) = (℩𝑥((𝐴 = ∅ ∧ 𝑥 = 0 ) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3705  dom cdm 4769  ℩cio 5330   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  seqcseq 10862  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587   Σ_gz cgzsu 13588

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-recs 6566  df-frec 6652  df-neg 8490  df-inn 9284  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gzsumval  13687