Theorem usgreqdrusgr

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
Hypotheses	isrusgr0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isrusgr0.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
Assertion	usgreqdrusgr	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) → 𝐺 RegUSGraph 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		isrusgr0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isrusgr0.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
3	1 2	isrusgr0	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ) → ( 𝐺 RegUSGraph 𝐾 ↔ ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
4	3	3adant3	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) → ( 𝐺 RegUSGraph 𝐾 ↔ ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
5	4	ibir	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) → 𝐺 RegUSGraph 𝐾 )