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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 29585 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0*) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))) | 
| 4 | 3 | 3adant3 1132 | . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 ‘cfv 6560 ℕ0*cxnn0 12601 Vtxcvtx 29014 USGraphcusgr 29167 VtxDegcvtxdg 29484 RegUSGraph crusgr 29575 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-iota 6513 df-fv 6568 df-rgr 29576 df-rusgr 29577 | 
| This theorem is referenced by: fusgrn0eqdrusgr 29589 frgrregorufrg 30346 | 
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