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Mirrors > Home > MPE Home > Th. List > usgreqdrusgr | Structured version Visualization version GIF version |
Description: If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
isrusgr0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isrusgr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
usgreqdrusgr | ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrusgr0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isrusgr0.d | . . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
3 | 1, 2 | isrusgr0 27347 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0*) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) |
4 | 3 | 3adant3 1128 | . 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 270 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5065 ‘cfv 6354 ℕ0*cxnn0 11966 Vtxcvtx 26780 USGraphcusgr 26933 VtxDegcvtxdg 27246 RegUSGraph crusgr 27337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-iota 6313 df-fv 6362 df-rgr 27338 df-rusgr 27339 |
This theorem is referenced by: fusgrn0eqdrusgr 27351 frgrregorufrg 28104 |
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