Theorem vtxdgoddnumeven 30146

Description: The number of vertices of odd degree is even in a finite pseudograph of finite size. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 22-Dec-2021.)

Hypotheses

Ref	Expression
finsumvtxdgeven.v	⊢ 𝑉 = (Vtx‘𝐺)
finsumvtxdgeven.i	⊢ 𝐼 = (iEdg‘𝐺)
finsumvtxdgeven.d	⊢ 𝐷 = (VtxDeg‘𝐺)

Assertion

Ref	Expression
vtxdgoddnumeven	⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉   𝑣,𝐷   𝑣,𝐼

Proof of Theorem vtxdgoddnumeven

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		finsumvtxdgeven.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		finsumvtxdgeven.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
3		finsumvtxdgeven.d	. . 3 ⊢ 𝐷 = (VtxDeg‘𝐺)
4	1, 2, 3	finsumvtxdgeven 30145	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤))
5		incom 4108	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
6		rabnc 4343	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) = ∅
7	5, 6	eqtri 2786	. . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ∅
8	7	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ∅)
9		rabxm 4342	. . . . . . 7 ⊢ 𝑉 = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
10	9	equncomi 4161	. . . . . 6 ⊢ 𝑉 = ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)})
11	10	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 𝑉 = ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}))
12		simp2 1155	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 𝑉 ∈ Fin)
13	3	fveq1i 6894	. . . . . 6 ⊢ (𝐷‘𝑤) = ((VtxDeg‘𝐺)‘𝑤)
14		dmfi 9335	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → dom 𝐼 ∈ Fin)
15	14	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → dom 𝐼 ∈ Fin)
16		eqid 2763	. . . . . . . . 9 ⊢ dom 𝐼 = dom 𝐼
17	1, 2, 16	vtxdgfisnn0 30068	. . . . . . . 8 ⊢ ((dom 𝐼 ∈ Fin ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
18	15, 17	sylan 592	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
19	18	nn0cnd 12679	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℂ)
20	13, 19	eqeltrid 2867	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → (𝐷‘𝑤) ∈ ℂ)
21	8, 11, 12, 20	fsumsplit 15919	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ 𝑉 (𝐷‘𝑤) = (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) + Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)))
22	21	breq2d 5117	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤) ↔ 2 ∥ (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) + Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))))
23		rabfi 9273	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
24	23	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
25		elrabi 3643	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} → 𝑤 ∈ 𝑉)
26	15, 25, 17	syl2an 608	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
27	26	nn0zd 12728	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
28	13, 27	eqeltrid 2867	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
29	24, 28	fsumzcl 15913	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
30	29	adantr 486	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
31		fveq2 6893	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑤 → (𝐷‘𝑣) = (𝐷‘𝑤))
32	31	breq2d 5117	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (2 ∥ (𝐷‘𝑣) ↔ 2 ∥ (𝐷‘𝑤)))
33	32	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (¬ 2 ∥ (𝐷‘𝑣) ↔ ¬ 2 ∥ (𝐷‘𝑤)))
34	33	elrab 3647	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ ¬ 2 ∥ (𝐷‘𝑤)))
35	34	simprbi 503	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} → ¬ 2 ∥ (𝐷‘𝑤))
36	35	adantl 487	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ¬ 2 ∥ (𝐷‘𝑤))
37	24, 28, 36	sumodd 16570	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) ↔ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)))
38	37	notbid 321	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) ↔ ¬ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)))
39	38	biimpa 482	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → ¬ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
40		rabfi 9273	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
41	40	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
42		elrabi 3643	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} → 𝑤 ∈ 𝑉)
43	15, 42, 17	syl2an 608	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
44	43	nn0zd 12728	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
45	13, 44	eqeltrid 2867	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
46	41, 45	fsumzcl 15913	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
47	46	adantr 486	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
48	32	elrab 3647	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ 2 ∥ (𝐷‘𝑤)))
49	48	simprbi 503	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} → 2 ∥ (𝐷‘𝑤))
50	49	adantl 487	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → 2 ∥ (𝐷‘𝑤))
51	41, 45, 50	sumeven 16569	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
52	51	adantr 486	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
53		opeo 16547	. . . . . 6 ⊢ (((Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ ∧ ¬ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)) ∧ (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ ∧ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))) → ¬ 2 ∥ (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) + Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)))
54	30, 39, 47, 52, 53	syl22anc 852	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → ¬ 2 ∥ (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) + Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)))
55	54	ex 418	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ¬ 2 ∥ (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) + Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))))
56	55	con4d 116	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (2 ∥ (Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) + Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤)) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})))
57	22, 56	sylbid 243	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})))
58	4, 57	mpd 16	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  {crab 3415   ∩ cin 4091   ∪ cun 4151  ∅c0 4280   class class class wbr 5105  dom cdm 5654  ‘cfv 6546  (class class class)co 7427  Fincfn 8984  ℂcc 11208   + caddc 11213  2c2 12407  ℕ₀cn0 12616  ℤcz 12703  ♯chash 14484  Σcsu 15865   ∥ cdvds 16434  Vtxcvtx 29586  iEdgciedg 29587  UPGraphcupgr 29670  VtxDegcvtxdg 30058

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-inf2 9652  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287  ax-pre-sup 11288

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-disj 5073  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9444  df-oi 9514  df-dju 9992  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-div 11984  df-nn 12346  df-2 12415  df-3 12416  df-n0 12617  df-xnn0 12690  df-z 12704  df-uz 12976  df-rp 13131  df-xadd 13252  df-fz 13650  df-fzo 13799  df-seq 14155  df-exp 14215  df-hash 14485  df-cj 15278  df-re 15279  df-im 15280  df-sqrt 15414  df-abs 15415  df-clim 15667  df-sum 15866  df-dvds 16435  df-vtx 29588  df-iedg 29589  df-edg 29638  df-uhgr 29648  df-upgr 29672  df-vtxdg 30059

This theorem is used by:  fusgrvtxdgonume  30147