Theorem usgreqdrusgr 29294

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* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)

Distinct variable groups:   𝑣,𝐺   𝑣,𝐾

Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)

Proof of Theorem usgreqdrusgr

Step	Hyp	Ref	Expression
1		isrusgr0.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		isrusgr0.d	. . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺)
3	1, 2	isrusgr0 29292	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0*) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
4	3	3adant3 1129	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
5	4	ibir 268	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053   class class class wbr 5138  ‘cfv 6533  ℕ₀^*cxnn0 12541  Vtxcvtx 28725  USGraphcusgr 28878  VtxDegcvtxdg 29191   RegUSGraph crusgr 29282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-iota 6485  df-fv 6541  df-rgr 29283  df-rusgr 29284

This theorem is referenced by:  fusgrn0eqdrusgr  29296  frgrregorufrg  30048