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Theorem cusgr3cyclex 35141
Description: Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.)
Hypothesis
Ref Expression
cusgr3cyclex.1 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
cusgr3cyclex ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   𝑉(𝑓,𝑝)

Proof of Theorem cusgr3cyclex
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3anass 1095 . . . . . . 7 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
21bianass 642 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) ↔ ((𝐺 ∈ ComplUSGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)))
3 cusgrusgr 29436 . . . . . . . . 9 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
4 usgrumgr 29198 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
53, 4syl 17 . . . . . . . 8 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ UMGraph)
6 3simpc 1151 . . . . . . . . . . . . 13 ((𝑎𝑉𝑏𝑉𝑐𝑉) → (𝑏𝑉𝑐𝑉))
76ancli 548 . . . . . . . . . . . 12 ((𝑎𝑉𝑏𝑉𝑐𝑉) → ((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑏𝑉𝑐𝑉)))
8 df-3an 1089 . . . . . . . . . . . . 13 ((𝑎𝑏𝑎𝑐𝑏𝑐) ↔ ((𝑎𝑏𝑎𝑐) ∧ 𝑏𝑐))
98biimpi 216 . . . . . . . . . . . 12 ((𝑎𝑏𝑎𝑐𝑏𝑐) → ((𝑎𝑏𝑎𝑐) ∧ 𝑏𝑐))
10 an32 646 . . . . . . . . . . . . . . 15 ((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑎𝑏𝑎𝑐)) ↔ (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ (𝑏𝑉𝑐𝑉)))
1110anbi1i 624 . . . . . . . . . . . . . 14 (((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑎𝑏𝑎𝑐)) ∧ 𝑏𝑐) ↔ ((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑏𝑐))
12 anass 468 . . . . . . . . . . . . . 14 (((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑏𝑐) ↔ (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ ((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐)))
1311, 12sylbb 219 . . . . . . . . . . . . 13 (((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ (𝑎𝑏𝑎𝑐)) ∧ 𝑏𝑐) → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ ((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐)))
1413anasss 466 . . . . . . . . . . . 12 ((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ ((𝑎𝑏𝑎𝑐) ∧ 𝑏𝑐)) → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ ((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐)))
157, 9, 14syl2an 596 . . . . . . . . . . 11 (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ ((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐)))
16 anandi3 1102 . . . . . . . . . . . . . . 15 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ ((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑉𝑐𝑉)))
1716anbi1i 624 . . . . . . . . . . . . . 14 (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ↔ (((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑉𝑐𝑉)) ∧ (𝑎𝑏𝑎𝑐)))
18 an4 656 . . . . . . . . . . . . . 14 ((((𝑎𝑉𝑏𝑉) ∧ (𝑎𝑉𝑐𝑉)) ∧ (𝑎𝑏𝑎𝑐)) ↔ (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) ∧ ((𝑎𝑉𝑐𝑉) ∧ 𝑎𝑐)))
1917, 18sylbb 219 . . . . . . . . . . . . 13 (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) ∧ ((𝑎𝑉𝑐𝑉) ∧ 𝑎𝑐)))
20 df-3an 1089 . . . . . . . . . . . . . . 15 ((𝑎𝑉𝑏𝑉𝑎𝑏) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏))
21 cusgr3cyclex.1 . . . . . . . . . . . . . . . 16 𝑉 = (Vtx‘𝐺)
22 eqid 2737 . . . . . . . . . . . . . . . 16 (Edg‘𝐺) = (Edg‘𝐺)
2321, 22cusgredgex2 35128 . . . . . . . . . . . . . . 15 (𝐺 ∈ ComplUSGraph → ((𝑎𝑉𝑏𝑉𝑎𝑏) → {𝑎, 𝑏} ∈ (Edg‘𝐺)))
2420, 23biimtrrid 243 . . . . . . . . . . . . . 14 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) → {𝑎, 𝑏} ∈ (Edg‘𝐺)))
25 df-3an 1089 . . . . . . . . . . . . . . 15 ((𝑎𝑉𝑐𝑉𝑎𝑐) ↔ ((𝑎𝑉𝑐𝑉) ∧ 𝑎𝑐))
2621, 22cusgredgex2 35128 . . . . . . . . . . . . . . 15 (𝐺 ∈ ComplUSGraph → ((𝑎𝑉𝑐𝑉𝑎𝑐) → {𝑎, 𝑐} ∈ (Edg‘𝐺)))
2725, 26biimtrrid 243 . . . . . . . . . . . . . 14 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑐𝑉) ∧ 𝑎𝑐) → {𝑎, 𝑐} ∈ (Edg‘𝐺)))
2824, 27anim12d 609 . . . . . . . . . . . . 13 (𝐺 ∈ ComplUSGraph → ((((𝑎𝑉𝑏𝑉) ∧ 𝑎𝑏) ∧ ((𝑎𝑉𝑐𝑉) ∧ 𝑎𝑐)) → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺))))
2919, 28syl5 34 . . . . . . . . . . . 12 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺))))
30 df-3an 1089 . . . . . . . . . . . . 13 ((𝑏𝑉𝑐𝑉𝑏𝑐) ↔ ((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐))
3121, 22cusgredgex2 35128 . . . . . . . . . . . . 13 (𝐺 ∈ ComplUSGraph → ((𝑏𝑉𝑐𝑉𝑏𝑐) → {𝑏, 𝑐} ∈ (Edg‘𝐺)))
3230, 31biimtrrid 243 . . . . . . . . . . . 12 (𝐺 ∈ ComplUSGraph → (((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐) → {𝑏, 𝑐} ∈ (Edg‘𝐺)))
3329, 32anim12d 609 . . . . . . . . . . 11 (𝐺 ∈ ComplUSGraph → ((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐)) ∧ ((𝑏𝑉𝑐𝑉) ∧ 𝑏𝑐)) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺)) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
3415, 33syl5 34 . . . . . . . . . 10 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺)) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
35 3anan32 1097 . . . . . . . . . . 11 (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺)) ↔ (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺)) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))
36 prcom 4732 . . . . . . . . . . . . 13 {𝑎, 𝑐} = {𝑐, 𝑎}
3736eleq1i 2832 . . . . . . . . . . . 12 ({𝑎, 𝑐} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑎} ∈ (Edg‘𝐺))
38373anbi3i 1160 . . . . . . . . . . 11 (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺)) ↔ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))
3935, 38bitr3i 277 . . . . . . . . . 10 ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺)) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))
4034, 39imbitrdi 251 . . . . . . . . 9 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
41 pm5.3 572 . . . . . . . . 9 ((((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → ((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))))
4240, 41sylib 218 . . . . . . . 8 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → ((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))))
4321, 22umgr3cyclex 30202 . . . . . . . . . 10 ((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎))
44 3simpa 1149 . . . . . . . . . . 11 ((𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎) → (𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
45442eximi 1836 . . . . . . . . . 10 (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
4643, 45syl 17 . . . . . . . . 9 ((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
47463expib 1123 . . . . . . . 8 (𝐺 ∈ UMGraph → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
485, 42, 47sylsyld 61 . . . . . . 7 (𝐺 ∈ ComplUSGraph → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑎𝑏𝑎𝑐𝑏𝑐)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
4948expdimp 452 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) → ((𝑎𝑏𝑎𝑐𝑏𝑐) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
502, 49sylbir 235 . . . . 5 (((𝐺 ∈ ComplUSGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑎𝑏𝑎𝑐𝑏𝑐) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
5150reximdvva 3207 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑎𝑉) → (∃𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐) → ∃𝑏𝑉𝑐𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
5251reximdva 3168 . . 3 (𝐺 ∈ ComplUSGraph → (∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
53 id 22 . . . . . 6 (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
5453rexlimivw 3151 . . . . 5 (∃𝑐𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
5554rexlimivw 3151 . . . 4 (∃𝑏𝑉𝑐𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
5655rexlimivw 3151 . . 3 (∃𝑎𝑉𝑏𝑉𝑐𝑉𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
5752, 56syl6 35 . 2 (𝐺 ∈ ComplUSGraph → (∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3)))
5821fvexi 6920 . . 3 𝑉 ∈ V
59 hashgt23el 14463 . . 3 ((𝑉 ∈ V ∧ 2 < (♯‘𝑉)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐))
6058, 59mpan 690 . 2 (2 < (♯‘𝑉) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐))
6157, 60impel 505 1 ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  {cpr 4628   class class class wbr 5143  cfv 6561  0cc0 11155   < clt 11295  2c2 12321  3c3 12322  chash 14369  Vtxcvtx 29013  Edgcedg 29064  UMGraphcumgr 29098  USGraphcusgr 29166  ComplUSGraphccusgr 29427  Cyclesccycls 29805
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-xneg 13154  df-xadd 13155  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-s4 14889  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-usgr 29168  df-nbgr 29350  df-uvtx 29403  df-cplgr 29428  df-cusgr 29429  df-wlks 29617  df-trls 29710  df-pths 29734  df-cycls 29807
This theorem is referenced by:  cusgracyclt3v  35161
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