Theorem vtxdgoddnumeven 29589

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 29588	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤))
5		incom 4230	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
6		rabnc 4414	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) = ∅
7	5, 6	eqtri 2768	. . . . . 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 4413	. . . . . . 7 ⊢ 𝑉 = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
10	9	equncomi 4183	. . . . . 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 6921	. . . . . 6 ⊢ (𝐷‘𝑤) = ((VtxDeg‘𝐺)‘𝑤)
14		dmfi 9403	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → dom 𝐼 ∈ Fin)
15	14	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → dom 𝐼 ∈ Fin)
16		eqid 2740	. . . . . . . . 9 ⊢ dom 𝐼 = dom 𝐼
17	1, 2, 16	vtxdgfisnn0 29511	. . . . . . . 8 ⊢ ((dom 𝐼 ∈ Fin ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
18	15, 17	sylan 579	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
19	18	nn0cnd 12615	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℂ)
20	13, 19	eqeltrid 2848	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → (𝐷‘𝑤) ∈ ℂ)
21	8, 11, 12, 20	fsumsplit 15789	. . . 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 5178	. . 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 9331	. . . . . . . . 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 3703	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} → 𝑤 ∈ 𝑉)
26	15, 25, 17	syl2an 595	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
27	26	nn0zd 12665	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
28	13, 27	eqeltrid 2848	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
29	24, 28	fsumzcl 15783	. . . . . . 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 6920	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑤 → (𝐷‘𝑣) = (𝐷‘𝑤))
32	31	breq2d 5178	. . . . . . . . . . . . 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 3708	. . . . . . . . . . 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 16436	. . . . . . . 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 9331	. . . . . . . . 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 3703	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} → 𝑤 ∈ 𝑉)
43	15, 42, 17	syl2an 595	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℕ0)
44	43	nn0zd 12665	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
45	13, 44	eqeltrid 2848	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
46	41, 45	fsumzcl 15783	. . . . . . 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 3708	. . . . . . . . . 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 16435	. . . . . . 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 16413	. . . . . 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 1087   = wceq 1537   ∈ wcel 2108  {crab 3443   ∪ cun 3974   ∩ cin 3975  ∅c0 4352   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℂcc 11182   + caddc 11187  2c2 12348  ℕ₀cn0 12553  ℤcz 12639  ♯chash 14379  Σcsu 15734   ∥ cdvds 16302  Vtxcvtx 29031  iEdgciedg 29032  UPGraphcupgr 29115  VtxDegcvtxdg 29501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-rp 13058  df-xadd 13176  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-dvds 16303  df-vtx 29033  df-iedg 29034  df-edg 29083  df-uhgr 29093  df-upgr 29117  df-vtxdg 29502

This theorem is referenced by:  fusgrvtxdgonume  29590