Theorem vtxdgoddnumeven 27920

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 27919	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑤 ∈ 𝑉 (𝐷‘𝑤))
5		incom 4135	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
6		rabnc 4321	. . . . . . 7 ⊢ ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∩ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) = ∅
7	5, 6	eqtri 2766	. . . . . 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 4320	. . . . . . 7 ⊢ 𝑉 = ({𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∪ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})
10	9	equncomi 4089	. . . . . 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 1136	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 𝑉 ∈ Fin)
13	3	fveq1i 6775	. . . . . 6 ⊢ (𝐷‘𝑤) = ((VtxDeg‘𝐺)‘𝑤)
14		dmfi 9097	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → dom 𝐼 ∈ Fin)
15	14	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → dom 𝐼 ∈ Fin)
16		eqid 2738	. . . . . . . . 9 ⊢ dom 𝐼 = dom 𝐼
17	1, 2, 16	vtxdgfisnn0 27842	. . . . . . . 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 12295	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℂ)
20	13, 19	eqeltrid 2843	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ 𝑉) → (𝐷‘𝑤) ∈ ℂ)
21	8, 11, 12, 20	fsumsplit 15453	. . . 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 5086	. . 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 9044	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
24	23	3ad2ant2 1133	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
25		elrabi 3618	. . . . . . . . . . 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 12424	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
28	13, 27	eqeltrid 2843	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
29	24, 28	fsumzcl 15447	. . . . . . 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 6774	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑤 → (𝐷‘𝑣) = (𝐷‘𝑤))
32	31	breq2d 5086	. . . . . . . . . . . . 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 3624	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ ¬ 2 ∥ (𝐷‘𝑤)))
35	34	simprbi 497	. . . . . . . . . 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 16097	. . . . . . . 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 477	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ ¬ 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) → ¬ 2 ∥ Σ𝑤 ∈ {𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)} (𝐷‘𝑤))
40		rabfi 9044	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
41	40	3ad2ant2 1133	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ∈ Fin)
42		elrabi 3618	. . . . . . . . . . 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 12424	. . . . . . . . 9 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → ((VtxDeg‘𝐺)‘𝑤) ∈ ℤ)
45	13, 44	eqeltrid 2843	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) ∧ 𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)}) → (𝐷‘𝑤) ∈ ℤ)
46	41, 45	fsumzcl 15447	. . . . . . 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 3624	. . . . . . . . . 10 ⊢ (𝑤 ∈ {𝑣 ∈ 𝑉 ∣ 2 ∥ (𝐷‘𝑣)} ↔ (𝑤 ∈ 𝑉 ∧ 2 ∥ (𝐷‘𝑤)))
49	48	simprbi 497	. . . . . . . . 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 16096	. . . . . . 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 16074	. . . . . 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 836	. . . . 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 413	. . . 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 239	. 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 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068   ∪ cun 3885   ∩ cin 3886  ∅c0 4256   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℂcc 10869   + caddc 10874  2c2 12028  ℕ₀cn0 12233  ℤcz 12319  ♯chash 14044  Σcsu 15397   ∥ cdvds 15963  Vtxcvtx 27366  iEdgciedg 27367  UPGraphcupgr 27450  VtxDegcvtxdg 27832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-rp 12731  df-xadd 12849  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-dvds 15964  df-vtx 27368  df-iedg 27369  df-edg 27418  df-uhgr 27428  df-upgr 27452  df-vtxdg 27833

This theorem is referenced by:  fusgrvtxdgonume  27921