Theorem cusgrsize 40670

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		edgaval 25794	. . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	syl5eq 2656	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph → 𝐸 = ran (iEdg‘𝐺))
4	3	adantr 480	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐸 = ran (iEdg‘𝐺))
5	4	fveq2d 6107	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = (#‘ran (iEdg‘𝐺)))
6		cusgrsizeindb0.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	opeq1i 4343	. . . 4 ⊢ ⟨𝑉, (iEdg‘𝐺)⟩ = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
8		cusgrop 40660	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
9	7, 8	syl5eqel 2692	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → ⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
10		fvex 6113	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
11		fvex 6113	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
12		rabexg 4739	. . . . . 6 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} ∈ V)
13	12	resiexd 6385	. . . . 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 5272	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝐺) → ran 𝑒 = ran (iEdg‘𝐺))
16	15	fveq2d 6107	. . . . 5 ⊢ (𝑒 = (iEdg‘𝐺) → (#‘ran 𝑒) = (#‘ran (iEdg‘𝐺)))
17		fveq2 6103	. . . . . 6 ⊢ (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
18	17	oveq1d 6564	. . . . 5 ⊢ (𝑣 = 𝑉 → ((#‘𝑣)C2) = ((#‘𝑉)C2))
19	16, 18	eqeqan12rd 2628	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = (iEdg‘𝐺)) → ((#‘ran 𝑒) = ((#‘𝑣)C2) ↔ (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2)))
20		rneq 5272	. . . . . 6 ⊢ (𝑒 = 𝑓 → ran 𝑒 = ran 𝑓)
21	20	fveq2d 6107	. . . . 5 ⊢ (𝑒 = 𝑓 → (#‘ran 𝑒) = (#‘ran 𝑓))
22		fveq2 6103	. . . . . 6 ⊢ (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
23	22	oveq1d 6564	. . . . 5 ⊢ (𝑣 = 𝑤 → ((#‘𝑣)C2) = ((#‘𝑤)C2))
24	21, 23	eqeqan12rd 2628	. . . 4 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((#‘ran 𝑒) = ((#‘𝑣)C2) ↔ (#‘ran 𝑓) = ((#‘𝑤)C2)))
25		vex 3176	. . . . . . 7 ⊢ 𝑣 ∈ V
26		vex 3176	. . . . . . 7 ⊢ 𝑒 ∈ V
27		opvtxfv 25681	. . . . . . 7 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣)
28	25, 26, 27	mp2an 704	. . . . . 6 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
29	28	eqcomi 2619	. . . . 5 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
30		eqid 2610	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
31		eqid 2610	. . . . 5 ⊢ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
32		eqid 2610	. . . . 5 ⊢ ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩
33	29, 30, 31, 32	cusgrres 40664	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ ∈ ComplUSGraph)
34		rneq 5272	. . . . . . 7 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → ran 𝑓 = ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}))
35	34	fveq2d 6107	. . . . . 6 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → (#‘ran 𝑓) = (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
36	35	adantl 481	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (#‘ran 𝑓) = (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
37		fveq2 6103	. . . . . . 7 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
38	37	adantr 480	. . . . . 6 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
39	38	oveq1d 6564	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((#‘𝑤)C2) = ((#‘(𝑣 ∖ {𝑛}))C2))
40	36, 39	eqeq12d 2625	. . . 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)))
41		edgaopval 25795	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
42	25, 26, 41	mp2an 704	. . . . . . . 8 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒
43	42	a1i 11	. . . . . . 7 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
44	43	eqcomd 2616	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩))
45	44	fveq2d 6107	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘ran 𝑒) = (#‘(Edg‘⟨𝑣, 𝑒⟩)))
46		cusgrusgr 40641	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ USGraph )
47		usgruhgr 40413	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ USGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph )
48	46, 47	syl 17	. . . . . 6 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph )
49	29, 30	cusgrsizeindb0 40665	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ UHGraph ∧ (#‘𝑣) = 0) → (#‘(Edg‘⟨𝑣, 𝑒⟩)) = ((#‘𝑣)C2))
50	48, 49	sylan 487	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘(Edg‘⟨𝑣, 𝑒⟩)) = ((#‘𝑣)C2))
51	45, 50	eqtrd 2644	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘ran 𝑒) = ((#‘𝑣)C2))
52		rnresi 5398	. . . . . . . . . 10 ⊢ ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
53	52	fveq2i 6106	. . . . . . . . 9 ⊢ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = (#‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})
54	42	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
55	54	rabeqdv 3167	. . . . . . . . . 10 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐})
56	55	fveq2d 6107	. . . . . . . . 9 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (#‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}))
57	53, 56	syl5eq 2656	. . . . . . . 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}))
58	57	eqeq1d 2612	. . . . . . 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)))
59	58	biimpd 218	. . . . . 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)))
60	59	imdistani 722	. . . . 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)))
61	42	eqcomi 2619	. . . . . . 7 ⊢ ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩)
62		eqid 2610	. . . . . . 7 ⊢ {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}
63	29, 61, 62	cusgrsize2inds 40669	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ0 → ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) → ((#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘ran 𝑒) = ((#‘𝑣)C2))))
64	63	imp31 447	. . . . 5 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘ran 𝑒) = ((#‘𝑣)C2))
65	60, 64	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))
66	10, 14, 19, 24, 33, 40, 51, 65	opfi1ind 13139	. . 3 ⊢ ((⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2))
67	9, 66	sylan 487	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2))
68	5, 67	eqtrd 2644	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  ⟨cop 4131   I cid 4948  ran crn 5039   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℕ₀cn0 11169  Ccbc 12951  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792   USGraph cusgr 40379  ComplUSGraphccusgr 40553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-fac 12923  df-bc 12952  df-hash 12980  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-uvtxa 40556  df-cplgr 40557  df-cusgr 40558

This theorem is referenced by:  fusgrmaxsize  40680