Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vdgn1frgrv2 Structured version   Visualization version   GIF version

Theorem vdgn1frgrv2 41466
 Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)
Hypothesis
Ref Expression
vdn1frgrv2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
vdgn1frgrv2 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))

Proof of Theorem vdgn1frgrv2
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrusgr 41432 . . . . . 6 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
21anim1i 590 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
32adantr 480 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
4 vdn1frgrv2.v . . . . 5 𝑉 = (Vtx‘𝐺)
5 eqid 2610 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
6 eqid 2610 . . . . 5 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
7 eqid 2610 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
84, 5, 6, 7vtxdusgrval 40702 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
93, 8syl 17 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
10 eqid 2610 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
114, 103cyclfrgrrn2 41457 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
1211adantlr 747 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))))
13 preq1 4212 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
1413eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ {𝑁, 𝑏} ∈ (Edg‘𝐺)))
15 preq2 4213 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
1615eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ (Edg‘𝐺) ↔ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
1714, 163anbi13d 1393 . . . . . . . . . . . . . 14 (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)) ↔ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
1817anbi2d 736 . . . . . . . . . . . . 13 (𝑎 = 𝑁 → ((𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
19182rexbidv 3039 . . . . . . . . . . . 12 (𝑎 = 𝑁 → (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) ↔ ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))))
2019rspcva 3280 . . . . . . . . . . 11 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → ∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))))
211adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝐺 ∈ USGraph )
22 simplll 794 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝑏𝑐)
23 3simpb 1052 . . . . . . . . . . . . . . . . . 18 (({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
2423ad3antlr 763 . . . . . . . . . . . . . . . . 17 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺)))
255, 10usgr2edg1 40439 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ USGraph ∧ 𝑏𝑐) ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2621, 22, 24, 25syl21anc 1317 . . . . . . . . . . . . . . . 16 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2726a1d 25 . . . . . . . . . . . . . . 15 ((((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) ∧ 𝐺 ∈ FriendGraph ) → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2827ex 449 . . . . . . . . . . . . . 14 (((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) ∧ 𝑁𝑉) → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))
2928ex 449 . . . . . . . . . . . . 13 ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3029a1i 11 . . . . . . . . . . . 12 ((𝑏𝑉𝑐𝑉) → ((𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
3130rexlimivv 3018 . . . . . . . . . . 11 (∃𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑁, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑁} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3220, 31syl 17 . . . . . . . . . 10 ((𝑁𝑉 ∧ ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺)))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3332ex 449 . . . . . . . . 9 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝑁𝑉 → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))))))
3433pm2.43a 52 . . . . . . . 8 (𝑁𝑉 → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (1 < (#‘𝑉) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3534com24 93 . . . . . . 7 (𝑁𝑉 → (1 < (#‘𝑉) → (𝐺 ∈ FriendGraph → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3635com3r 85 . . . . . 6 (𝐺 ∈ FriendGraph → (𝑁𝑉 → (1 < (#‘𝑉) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))))
3736imp31 447 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑎} ∈ (Edg‘𝐺))) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
3812, 37mpd 15 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
39 fvex 6113 . . . . . . . . 9 (iEdg‘𝐺) ∈ V
4039dmex 6991 . . . . . . . 8 dom (iEdg‘𝐺) ∈ V
4140a1i 11 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → dom (iEdg‘𝐺) ∈ V)
42 rabexg 4739 . . . . . . 7 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
43 hash1snb 13068 . . . . . . 7 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
4441, 42, 433syl 18 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖}))
45 reusn 4206 . . . . . 6 (∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ∃𝑖{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑖})
4644, 45syl6bbr 277 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1 ↔ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
4746necon3abid 2818 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1 ↔ ¬ ∃!𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
4838, 47mpbird 246 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 1)
499, 48eqnetrd 2849 . 2 (((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) ∧ 1 < (#‘𝑉)) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1)
5049ex 449 1 ((𝐺 ∈ FriendGraph ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((VtxDeg‘𝐺)‘𝑁) ≠ 1))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  1c1 9816   < clt 9953  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   USGraph cusgr 40379  VtxDegcvtxdg 40681   FriendGraph cfrgr 41428 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-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-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-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-vtxdg 40682  df-frgr 41429 This theorem is referenced by:  vdgn1frgrv3  41467  vdgfrgrgt2  41468
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