Theorem cusgrsize 29388

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 28982	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2753	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	a1i 11	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐸 = ran (iEdg‘𝐺))
5	4	fveq2d 6864	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = (♯‘ran (iEdg‘𝐺)))
6		cusgrsizeindb0.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	opeq1i 4842	. . . 4 ⊢ ⟨𝑉, (iEdg‘𝐺)⟩ = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
8		cusgrop 29371	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
9	7, 8	eqeltrid 2833	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → ⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
10		fvex 6873	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
11		fvex 6873	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
12		rabexg 5294	. . . . . 6 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} ∈ V)
13	12	resiexd 7192	. . . . 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 5902	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝐺) → ran 𝑒 = ran (iEdg‘𝐺))
16	15	fveq2d 6864	. . . . 5 ⊢ (𝑒 = (iEdg‘𝐺) → (♯‘ran 𝑒) = (♯‘ran (iEdg‘𝐺)))
17		fveq2 6860	. . . . . 6 ⊢ (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
18	17	oveq1d 7404	. . . . 5 ⊢ (𝑣 = 𝑉 → ((♯‘𝑣)C2) = ((♯‘𝑉)C2))
19	16, 18	eqeqan12rd 2745	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = (iEdg‘𝐺)) → ((♯‘ran 𝑒) = ((♯‘𝑣)C2) ↔ (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2)))
20		rneq 5902	. . . . . 6 ⊢ (𝑒 = 𝑓 → ran 𝑒 = ran 𝑓)
21	20	fveq2d 6864	. . . . 5 ⊢ (𝑒 = 𝑓 → (♯‘ran 𝑒) = (♯‘ran 𝑓))
22		fveq2 6860	. . . . . 6 ⊢ (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
23	22	oveq1d 7404	. . . . 5 ⊢ (𝑣 = 𝑤 → ((♯‘𝑣)C2) = ((♯‘𝑤)C2))
24	21, 23	eqeqan12rd 2745	. . . 4 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((♯‘ran 𝑒) = ((♯‘𝑣)C2) ↔ (♯‘ran 𝑓) = ((♯‘𝑤)C2)))
25		vex 3454	. . . . . . 7 ⊢ 𝑣 ∈ V
26		vex 3454	. . . . . . 7 ⊢ 𝑒 ∈ V
27	25, 26	opvtxfvi 28942	. . . . . 6 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
28	27	eqcomi 2739	. . . . 5 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
29		eqid 2730	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
30		eqid 2730	. . . . 5 ⊢ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
31		eqid 2730	. . . . 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 29382	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ ∈ ComplUSGraph)
33		rneq 5902	. . . . . . 7 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → ran 𝑓 = ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}))
34	33	fveq2d 6864	. . . . . 6 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → (♯‘ran 𝑓) = (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
35	34	adantl 481	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (♯‘ran 𝑓) = (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
36		fveq2 6860	. . . . . . 7 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
37	36	adantr 480	. . . . . 6 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
38	37	oveq1d 7404	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((♯‘𝑤)C2) = ((♯‘(𝑣 ∖ {𝑛}))C2))
39	35, 38	eqeq12d 2746	. . . 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 28984	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
41	40	el2v 3457	. . . . . . . 8 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒
42	41	a1i 11	. . . . . . 7 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
43	42	eqcomd 2736	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩))
44	43	fveq2d 6864	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘ran 𝑒) = (♯‘(Edg‘⟨𝑣, 𝑒⟩)))
45		cusgrusgr 29352	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ USGraph)
46		usgruhgr 29119	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ USGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph)
47	45, 46	syl 17	. . . . . 6 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph)
48	28, 29	cusgrsizeindb0 29383	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ UHGraph ∧ (♯‘𝑣) = 0) → (♯‘(Edg‘⟨𝑣, 𝑒⟩)) = ((♯‘𝑣)C2))
49	47, 48	sylan 580	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘(Edg‘⟨𝑣, 𝑒⟩)) = ((♯‘𝑣)C2))
50	44, 49	eqtrd 2765	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘ran 𝑒) = ((♯‘𝑣)C2))
51		rnresi 6048	. . . . . . . . . 10 ⊢ ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
52	51	fveq2i 6863	. . . . . . . . 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 3424	. . . . . . . . . 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 6864	. . . . . . . . 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 2777	. . . . . . . 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 2732	. . . . . . 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 229	. . . . . 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 568	. . . . 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 2739	. . . . . . 7 ⊢ ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩)
61		eqid 2730	. . . . . . 7 ⊢ {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}
62	28, 60, 61	cusgrsize2inds 29387	. . . . . 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 417	. . . . 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 14483	. . 3 ⊢ ((⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2))
66	9, 65	sylan 580	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2))
67	5, 66	eqtrd 2765	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∉ wnel 3030  {crab 3408  Vcvv 3450   ∖ cdif 3913  {csn 4591  ⟨cop 4597   I cid 5534  ran crn 5641   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389  Fincfn 8920  0cc0 11074  1c1 11075   + caddc 11077  2c2 12242  ℕ₀cn0 12448  Ccbc 14273  ♯chash 14301  Vtxcvtx 28929  iEdgciedg 28930  Edgcedg 28980  UHGraphcuhgr 28989  USGraphcusgr 29082  ComplUSGraphccusgr 29343

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-oadd 8440  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-dju 9860  df-card 9898  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12188  df-2 12250  df-n0 12449  df-xnn0 12522  df-z 12536  df-uz 12800  df-rp 12958  df-fz 13475  df-seq 13973  df-fac 14245  df-bc 14274  df-hash 14302  df-vtx 28931  df-iedg 28932  df-edg 28981  df-uhgr 28991  df-upgr 29015  df-umgr 29016  df-uspgr 29083  df-usgr 29084  df-fusgr 29250  df-nbgr 29266  df-uvtx 29319  df-cplgr 29344  df-cusgr 29345

This theorem is referenced by:  fusgrmaxsize  29398