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Theorem rusgrprop0 26367
Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrusgr0.v 𝑉 = (Vtx‘𝐺)
isrusgr0.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
rusgrprop0 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)

Proof of Theorem rusgrprop0
StepHypRef Expression
1 rusgrprop 26362 . 2 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
2 isrusgr0.v . . . . 5 𝑉 = (Vtx‘𝐺)
3 isrusgr0.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
42, 3rgrprop 26360 . . . 4 (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
54anim2i 592 . . 3 ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
6 3anass 1040 . . 3 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
75, 6sylibr 224 . 2 ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
81, 7syl 17 1 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908   class class class wbr 4623  cfv 5857  0*cxnn0 11323  Vtxcvtx 25808   USGraph cusgr 25971  VtxDegcvtxdg 26282   RegGraph crgr 26355   RegUSGraph crusgr 26356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-iota 5820  df-fv 5865  df-rgr 26357  df-rusgr 26358
This theorem is referenced by:  frusgrnn0  26371  cusgrm1rusgr  26382  rusgrpropnb  26383  frgrreg  27140  frgrregord013  27141
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