Theorem vtxdgoddnumeven 29499

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 29498	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤))
5		incom 4160	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
6		rabnc 4342	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) = ∅
7	5, 6	eqtri 2752	. . . . . 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 4341	. . . . . . 7 ⊢ 𝑉 = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
10	9	equncomi 4111	. . . . . 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 1137	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 𝑉 ∈ Fin)
13	3	fveq1i 6823	. . . . . 6 ⊢ (𝐷‘𝑤) = ((VtxDeg‘𝐺)‘𝑤)
14		dmfi 9225	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → dom 𝐼 ∈ Fin)
15	14	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → dom 𝐼 ∈ Fin)
16		eqid 2729	. . . . . . . . 9 ⊢ dom 𝐼 = dom 𝐼
17	1, 2, 16	vtxdgfisnn0 29421	. . . . . . . 8 ⊢ ((dom 𝐼 ∈ Fin ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
18	15, 17	sylan 580	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
19	18	nn0cnd 12447	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℂ)
20	13, 19	eqeltrid 2832	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → (𝐷‘𝑤) ∈ ℂ)
21	8, 11, 12, 20	fsumsplit 15648	. . . 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 5104	. . 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 9160	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
24	23	3ad2ant2 1134	. . . . . . . 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 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
27	26	nn0zd 12497	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
28	13, 27	eqeltrid 2832	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
29	24, 28	fsumzcl 15642	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
30	29	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
31		fveq2 6822	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑤 → (𝐷‘𝑣) = (𝐷‘𝑤))
32	31	breq2d 5104	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (2 ∥ (𝐷‘𝑣) ↔ 2 ∥ (𝐷‘𝑤)))
33	32	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (¬ 2 ∥ (𝐷‘𝑣) ↔ ¬ 2 ∥ (𝐷‘𝑤)))
34	33	elrab 3648	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ ¬ 2 ∥ (𝐷‘𝑤)))
35	34	simprbi 496	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} → ¬ 2 ∥ (𝐷‘𝑤))
36	35	adantl 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ¬ 2 ∥ (𝐷‘𝑤))
37	24, 28, 36	sumodd 16299	. . . . . . . 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 318	. . . . . . 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 476	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → ¬ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
40		rabfi 9160	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
41	40	3ad2ant2 1134	. . . . . . . 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 596	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
44	43	nn0zd 12497	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
45	13, 44	eqeltrid 2832	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
46	41, 45	fsumzcl 15642	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
47	46	adantr 480	. . . . . 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 3648	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ 2 ∥ (𝐷‘𝑤)))
49	48	simprbi 496	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} → 2 ∥ (𝐷‘𝑤))
50	49	adantl 481	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → 2 ∥ (𝐷‘𝑤))
51	41, 45, 50	sumeven 16298	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
52	51	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
53		opeo 16276	. . . . . 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 838	. . . . 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 412	. . . 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 115	. . 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 240	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → (2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})))
58	4, 57	mpd 15	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3394   ∪ cun 3901   ∩ cin 3902  ∅c0 4284   class class class wbr 5092  dom cdm 5619  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  ℂcc 11007   + caddc 11012  2c2 12183  ℕ₀cn0 12384  ℤcz 12471  ♯chash 14237  Σcsu 15593   ∥ cdvds 16163  Vtxcvtx 28941  iEdgciedg 28942  UPGraphcupgr 29025  VtxDegcvtxdg 29411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-oi 9402  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-rp 12894  df-xadd 13015  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594  df-dvds 16164  df-vtx 28943  df-iedg 28944  df-edg 28993  df-uhgr 29003  df-upgr 29027  df-vtxdg 29412

This theorem is referenced by:  fusgrvtxdgonume  29500