Theorem vtxdgoddnumeven 26449

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 26448	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤))
5		incom 3805	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
6		rabnc 3962	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) = ∅
7	5, 6	eqtri 2644	. . . . . 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 3961	. . . . . . 7 ⊢ 𝑉 = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
10	9	equncomi 3759	. . . . . 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 1062	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 𝑉 ∈ Fin)
13	3	fveq1i 6192	. . . . . 6 ⊢ (𝐷‘𝑤) = ((VtxDeg‘𝐺)‘𝑤)
14		dmfi 8244	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → dom 𝐼 ∈ Fin)
15	14	3ad2ant3 1084	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → dom 𝐼 ∈ Fin)
16		eqid 2622	. . . . . . . . 9 ⊢ dom 𝐼 = dom 𝐼
17	1, 2, 16	vtxdgfisnn0 26371	. . . . . . . 8 ⊢ ((dom 𝐼 ∈ Fin ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
18	15, 17	sylan 488	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
19	18	nn0cnd 11353	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℂ)
20	13, 19	syl5eqel 2705	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → (𝐷‘𝑤) ∈ ℂ)
21	8, 11, 12, 20	fsumsplit 14471	. . . 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 4665	. . 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 8185	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
24	23	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
25		elrabi 3359	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} → 𝑤 ∈ 𝑉)
26	15, 25, 17	syl2an 494	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
27	26	nn0zd 11480	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
28	13, 27	syl5eqel 2705	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
29	24, 28	fsumzcl 14466	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
30	29	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (#‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
31		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑤 → (𝐷‘𝑣) = (𝐷‘𝑤))
32	31	breq2d 4665	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (2 ∥ (𝐷‘𝑣) ↔ 2 ∥ (𝐷‘𝑤)))
33	32	notbid 308	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (¬ 2 ∥ (𝐷‘𝑣) ↔ ¬ 2 ∥ (𝐷‘𝑤)))
34	33	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ ¬ 2 ∥ (𝐷‘𝑤)))
35	34	simprbi 480	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} → ¬ 2 ∥ (𝐷‘𝑤))
36	35	adantl 482	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ¬ 2 ∥ (𝐷‘𝑤))
37	24, 28, 36	sumodd 15111	. . . . . . . 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 308	. . . . . . 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 501	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (#‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → ¬ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
40		rabfi 8185	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
41	40	3ad2ant2 1083	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
42		elrabi 3359	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} → 𝑤 ∈ 𝑉)
43	15, 42, 17	syl2an 494	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
44	43	nn0zd 11480	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
45	13, 44	syl5eqel 2705	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
46	41, 45	fsumzcl 14466	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤) ∈ ℤ)
47	46	adantr 481	. . . . . 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 3363	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ 2 ∥ (𝐷‘𝑤)))
49	48	simprbi 480	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} → 2 ∥ (𝐷‘𝑤))
50	49	adantl 482	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → 2 ∥ (𝐷‘𝑤))
51	41, 45, 50	sumeven 15110	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
52	51	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (#‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
53		opeo 15089	. . . . . 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 1327	. . . . 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 450	. . . 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 114	. . 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 230	. 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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916   ∪ cun 3572   ∩ cin 3573  ∅c0 3915   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934   + caddc 9939  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  #chash 13117  Σcsu 14416   ∥ cdvds 14983  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975  VtxDegcvtxdg 26361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-vtxdg 26362

This theorem is referenced by:  fusgrvtxdgonume  26450