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Theorem isrusgr0 27360
 Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrusgr0.v 𝑉 = (Vtx‘𝐺)
isrusgr0.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
isrusgr0 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)   𝑊(𝑣)   𝑍(𝑣)

Proof of Theorem isrusgr0
StepHypRef Expression
1 isrusgr 27355 . 2 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
2 isrusgr0.v . . . . 5 𝑉 = (Vtx‘𝐺)
3 isrusgr0.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
42, 3isrgr 27353 . . . 4 ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
54anbi2d 631 . . 3 ((𝐺𝑊𝐾𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))))
6 3anass 1092 . . 3 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
75, 6syl6bbr 292 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
81, 7bitrd 282 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ∀wral 3109   class class class wbr 5033  ‘cfv 6328  ℕ0*cxnn0 11959  Vtxcvtx 26793  USGraphcusgr 26946  VtxDegcvtxdg 27259   RegGraph crgr 27349   RegUSGraph crusgr 27350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-iota 6287  df-fv 6336  df-rgr 27351  df-rusgr 27352 This theorem is referenced by:  usgreqdrusgr  27362  cusgrrusgr  27375  rgrusgrprc  27383  rusgrprc  27384
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