Theorem cusgrsize 26759

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 26354	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2849	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	a1i 11	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐸 = ran (iEdg‘𝐺))
5	4	fveq2d 6441	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = (♯‘ran (iEdg‘𝐺)))
6		cusgrsizeindb0.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	opeq1i 4628	. . . 4 ⊢ ⟨𝑉, (iEdg‘𝐺)⟩ = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
8		cusgrop 26743	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
9	7, 8	syl5eqel 2910	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → ⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
10		fvex 6450	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
11		fvex 6450	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
12		rabexg 5038	. . . . . 6 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} ∈ V)
13	12	resiexd 6741	. . . . 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 5587	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝐺) → ran 𝑒 = ran (iEdg‘𝐺))
16	15	fveq2d 6441	. . . . 5 ⊢ (𝑒 = (iEdg‘𝐺) → (♯‘ran 𝑒) = (♯‘ran (iEdg‘𝐺)))
17		fveq2 6437	. . . . . 6 ⊢ (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
18	17	oveq1d 6925	. . . . 5 ⊢ (𝑣 = 𝑉 → ((♯‘𝑣)C2) = ((♯‘𝑉)C2))
19	16, 18	eqeqan12rd 2843	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = (iEdg‘𝐺)) → ((♯‘ran 𝑒) = ((♯‘𝑣)C2) ↔ (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2)))
20		rneq 5587	. . . . . 6 ⊢ (𝑒 = 𝑓 → ran 𝑒 = ran 𝑓)
21	20	fveq2d 6441	. . . . 5 ⊢ (𝑒 = 𝑓 → (♯‘ran 𝑒) = (♯‘ran 𝑓))
22		fveq2 6437	. . . . . 6 ⊢ (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
23	22	oveq1d 6925	. . . . 5 ⊢ (𝑣 = 𝑤 → ((♯‘𝑣)C2) = ((♯‘𝑤)C2))
24	21, 23	eqeqan12rd 2843	. . . 4 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((♯‘ran 𝑒) = ((♯‘𝑣)C2) ↔ (♯‘ran 𝑓) = ((♯‘𝑤)C2)))
25		vex 3417	. . . . . . 7 ⊢ 𝑣 ∈ V
26		vex 3417	. . . . . . 7 ⊢ 𝑒 ∈ V
27	25, 26	opvtxfvi 26314	. . . . . 6 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
28	27	eqcomi 2834	. . . . 5 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
29		eqid 2825	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
30		eqid 2825	. . . . 5 ⊢ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
31		eqid 2825	. . . . 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 26753	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ ∈ ComplUSGraph)
33		rneq 5587	. . . . . . 7 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → ran 𝑓 = ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}))
34	33	fveq2d 6441	. . . . . 6 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → (♯‘ran 𝑓) = (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
35	34	adantl 475	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (♯‘ran 𝑓) = (♯‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
36		fveq2 6437	. . . . . . 7 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
37	36	adantr 474	. . . . . 6 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (♯‘𝑤) = (♯‘(𝑣 ∖ {𝑛})))
38	37	oveq1d 6925	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((♯‘𝑤)C2) = ((♯‘(𝑣 ∖ {𝑛}))C2))
39	35, 38	eqeq12d 2840	. . . 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 26356	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
41	25, 26, 40	mp2an 683	. . . . . . . 8 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒
42	41	a1i 11	. . . . . . 7 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
43	42	eqcomd 2831	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩))
44	43	fveq2d 6441	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘ran 𝑒) = (♯‘(Edg‘⟨𝑣, 𝑒⟩)))
45		cusgrusgr 26724	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ USGraph)
46		usgruhgr 26489	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ USGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph)
47	45, 46	syl 17	. . . . . 6 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph)
48	28, 29	cusgrsizeindb0 26754	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ UHGraph ∧ (♯‘𝑣) = 0) → (♯‘(Edg‘⟨𝑣, 𝑒⟩)) = ((♯‘𝑣)C2))
49	47, 48	sylan 575	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘(Edg‘⟨𝑣, 𝑒⟩)) = ((♯‘𝑣)C2))
50	44, 49	eqtrd 2861	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (♯‘𝑣) = 0) → (♯‘ran 𝑒) = ((♯‘𝑣)C2))
51		rnresi 5724	. . . . . . . . . 10 ⊢ ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
52	51	fveq2i 6440	. . . . . . . . 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 3407	. . . . . . . . . 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 6441	. . . . . . . . 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	syl5eq 2873	. . . . . . . 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 2827	. . . . . . 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 221	. . . . . 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 564	. . . . 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 2834	. . . . . . 7 ⊢ ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩)
61		eqid 2825	. . . . . . 7 ⊢ {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}
62	28, 60, 61	cusgrsize2inds 26758	. . . . . 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 410	. . . . 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 13580	. . 3 ⊢ ((⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2))
66	9, 65	sylan 575	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘ran (iEdg‘𝐺)) = ((♯‘𝑉)C2))
67	5, 66	eqtrd 2861	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (♯‘𝐸) = ((♯‘𝑉)C2))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ∉ wnel 3102  {crab 3121  Vcvv 3414   ∖ cdif 3795  {csn 4399  ⟨cop 4405   I cid 5251  ran crn 5347   ↾ cres 5348  ‘cfv 6127  (class class class)co 6910  Fincfn 8228  0cc0 10259  1c1 10260   + caddc 10262  2c2 11413  ℕ₀cn0 11625  Ccbc 13389  ♯chash 13417  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352  UHGraphcuhgr 26361  USGraphcusgr 26455  ComplUSGraphccusgr 26715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-rp 12120  df-fz 12627  df-seq 13103  df-fac 13361  df-bc 13390  df-hash 13418  df-vtx 26303  df-iedg 26304  df-edg 26353  df-uhgr 26363  df-upgr 26387  df-umgr 26388  df-uspgr 26456  df-usgr 26457  df-fusgr 26621  df-nbgr 26637  df-uvtx 26691  df-cplgr 26716  df-cusgr 26717

This theorem is referenced by:  fusgrmaxsize  26769