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Theorem usgreqdrusgr 29587
Description: If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrusgr0.v 𝑉 = (Vtx‘𝐺)
isrusgr0.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
usgreqdrusgr ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)

Proof of Theorem usgreqdrusgr
StepHypRef Expression
1 isrusgr0.v . . . 4 𝑉 = (Vtx‘𝐺)
2 isrusgr0.d . . . 4 𝐷 = (VtxDeg‘𝐺)
31, 2isrusgr0 29585 . . 3 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0*) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
433adant3 1132 . 2 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
54ibir 268 1 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142  cfv 6560  0*cxnn0 12601  Vtxcvtx 29014  USGraphcusgr 29167  VtxDegcvtxdg 29484   RegUSGraph crusgr 29575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-iota 6513  df-fv 6568  df-rgr 29576  df-rusgr 29577
This theorem is referenced by:  fusgrn0eqdrusgr  29589  frgrregorufrg  30346
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