Theorem cusgrsize 28691

Description: The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ₀) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)

Hypotheses

Ref	Expression
cusgrsizeindb0.v	⊢ 𝑉 = (Vtx‘𝐺)
cusgrsizeindb0.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
cusgrsize	⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2))

Proof of Theorem cusgrsize

Dummy variables 𝑒 𝑓 𝑛 𝑣 𝑐 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrsizeindb0.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2		edgval 28289	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2761	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	a1i 11	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐸 = ran (iEdg‘𝐺))
5	4	fveq2d 6892	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = (♯‘ran (iEdg‘𝐺)))
6		cusgrsizeindb0.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	opeq1i 4875	. . . 4 ⊢ ⟨𝑉, (iEdg‘𝐺)⟩ = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
8		cusgrop 28675	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
9	7, 8	eqeltrid 2838	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → ⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
10		fvex 6901	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
11		fvex 6901	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
12		rabexg 5330	. . . . . 6 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} ∈ V)
13	12	resiexd 7213	. . . . 5 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) ∈ V)
14	11, 13	ax-mp 5	. . . 4 ⊢ ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) ∈ V
15		rneq 5933	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝐺) → ran 𝑒 = ran (iEdg‘𝐺))
16	15	fveq2d 6892	. . . . 5 ⊢ (𝑒 = (iEdg‘𝐺) → (♯‘ran 𝑒) = (♯‘ran (iEdg‘𝐺)))
17		fveq2 6888	. . . . . 6 ⊢ (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
18	17	oveq1d 7419	. . . . 5 ⊢ (𝑣 = 𝑉 → ((♯‘𝑣)C2) = ((♯‘𝑉)C2))
19	16, 18	eqeqan12rd 2748	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = (iEdg‘𝐺)) → ((♯‘ran 𝑒) = ((♯‘𝑣)C2) ↔ (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2)))
20		rneq 5933	. . . . . 6 ⊢ (𝑒 = 𝑓 → ran 𝑒 = ran 𝑓)
21	20	fveq2d 6892	. . . . 5 ⊢ (𝑒 = 𝑓 → (♯‘ran 𝑒) = (♯‘ran 𝑓))
22		fveq2 6888	. . . . . 6 ⊢ (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
23	22	oveq1d 7419	. . . . 5 ⊢ (𝑣 = 𝑤 → ((♯‘𝑣)C2) = ((♯‘𝑤)C2))
24	21, 23	eqeqan12rd 2748	. . . 4 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((♯‘ran 𝑒) = ((♯‘𝑣)C2) ↔ (♯‘ran 𝑓) = ((♯‘𝑤)C2)))
25		vex 3479	. . . . . . 7 ⊢ 𝑣 ∈ V
26		vex 3479	. . . . . . 7 ⊢ 𝑒 ∈ V
27	25, 26	opvtxfvi 28249	. . . . . 6 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
28	27	eqcomi 2742	. . . . 5 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
29		eqid 2733	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
30		eqid 2733	. . . . 5 ⊢ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
31		eqid 2733	. . . . 5 ⊢ ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩
32	28, 29, 30, 31	cusgrres 28685	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ ∈ ComplUSGraph)
33		rneq 5933	. . . . . . 7 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → ran 𝑓 = ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}))
34	33	fveq2d 6892	. . . . . 6 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → (♯‘ran 𝑓) = (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
35	34	adantl 483	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (♯‘ran 𝑓) = (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
36		fveq2 6888	. . . . . . 7 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
37	36	adantr 482	. . . . . 6 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
38	37	oveq1d 7419	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((♯‘𝑤)C2) = ((♯‘(𝑣 ∖ {𝑛}))C2))
39	35, 38	eqeq12d 2749	. . . 4 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((♯‘ran 𝑓) = ((♯‘𝑤)C2) ↔ (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((♯‘(𝑣 ∖ {𝑛}))C2)))
40		edgopval 28291	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
41	40	el2v 3483	. . . . . . . 8 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒
42	41	a1i 11	. . . . . . 7 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
43	42	eqcomd 2739	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩))
44	43	fveq2d 6892	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘ran 𝑒) = (♯‘(Edg‘⟨𝑣, 𝑒⟩)))
45		cusgrusgr 28656	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ USGraph)
46		usgruhgr 28423	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ USGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph)
47	45, 46	syl 17	. . . . . 6 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph)
48	28, 29	cusgrsizeindb0 28686	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ UHGraph ∧ (♯‘𝑣) = 0) → (♯‘(Edg‘⟨𝑣, 𝑒⟩)) = ((♯‘𝑣)C2))
49	47, 48	sylan 581	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘(Edg‘⟨𝑣, 𝑒⟩)) = ((♯‘𝑣)C2))
50	44, 49	eqtrd 2773	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘ran 𝑒) = ((♯‘𝑣)C2))
51		rnresi 6071	. . . . . . . . . 10 ⊢ ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
52	51	fveq2i 6891	. . . . . . . . 9 ⊢ (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = (♯‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})
53	41	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
54	53	rabeqdv 3448	. . . . . . . . . 10 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐})
55	54	fveq2d 6892	. . . . . . . . 9 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (♯‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = (♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}))
56	52, 55	eqtrid 2785	. . . . . . . 8 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = (♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}))
57	56	eqeq1d 2735	. . . . . . 7 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((♯‘(𝑣 ∖ {𝑛}))C2) ↔ (♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((♯‘(𝑣 ∖ {𝑛}))C2)))
58	57	biimpd 228	. . . . . 6 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((♯‘(𝑣 ∖ {𝑛}))C2) → (♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((♯‘(𝑣 ∖ {𝑛}))C2)))
59	58	imdistani 570	. . . . 5 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((♯‘(𝑣 ∖ {𝑛}))C2)) → (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((♯‘(𝑣 ∖ {𝑛}))C2)))
60	41	eqcomi 2742	. . . . . . 7 ⊢ ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩)
61		eqid 2733	. . . . . . 7 ⊢ {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}
62	28, 60, 61	cusgrsize2inds 28690	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ0 → ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) → ((♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((♯‘(𝑣 ∖ {𝑛}))C2) → (♯‘ran 𝑒) = ((♯‘𝑣)C2))))
63	62	imp31 419	. . . . 5 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (♯‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((♯‘(𝑣 ∖ {𝑛}))C2)) → (♯‘ran 𝑒) = ((♯‘𝑣)C2))
64	59, 63	syl 17	. . . 4 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((♯‘(𝑣 ∖ {𝑛}))C2)) → (♯‘ran 𝑒) = ((♯‘𝑣)C2))
65	10, 14, 19, 24, 32, 39, 50, 64	opfi1ind 14459	. . 3 ⊢ ((⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2))
66	9, 65	sylan 581	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2))
67	5, 66	eqtrd 2773	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∉ wnel 3047  {crab 3433  Vcvv 3475   ∖ cdif 3944  {csn 4627  ⟨cop 4633   I cid 5572  ran crn 5676   ↾ cres 5677  ‘cfv 6540  (class class class)co 7404  Fincfn 8935  0cc0 11106  1c1 11107   + caddc 11109  2c2 12263  ℕ₀cn0 12468  Ccbc 14258  ♯chash 14286  Vtxcvtx 28236  iEdgciedg 28237  Edgcedg 28287  UHGraphcuhgr 28296  USGraphcusgr 28389  ComplUSGraphccusgr 28647

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-seq 13963  df-fac 14230  df-bc 14259  df-hash 14287  df-vtx 28238  df-iedg 28239  df-edg 28288  df-uhgr 28298  df-upgr 28322  df-umgr 28323  df-uspgr 28390  df-usgr 28391  df-fusgr 28554  df-nbgr 28570  df-uvtx 28623  df-cplgr 28648  df-cusgr 28649

This theorem is referenced by:  fusgrmaxsize  28701