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Theorem usgreqdrusgr 26699
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) → 𝐺RegUSGraph𝐾)
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 26697 . . 3 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0*) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
433adant3 1126 . 2 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
54ibir 257 1 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → 𝐺RegUSGraph𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4786  cfv 6031  0*cxnn0 11565  Vtxcvtx 26095  USGraphcusgr 26266  VtxDegcvtxdg 26596  RegUSGraphcrusgr 26687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-iota 5994  df-fv 6039  df-rgr 26688  df-rusgr 26689
This theorem is referenced by:  fusgrn0eqdrusgr  26701  frgrregorufrg  27508
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