Theorem rgrprop

Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018) (Revised by AV, 26-Dec-2020)

	Ref	Expression
Hypotheses	isrgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isrgr.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
Assertion	rgrprop	⊢ ( 𝐺 RegGraph 𝐾 → ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		isrgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isrgr.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
3		df-rgr	⊢ RegGraph = { ⟨ 𝑔 , 𝑘 ⟩ ∣ ( 𝑘 ∈ ℕ0* ∧ ∀ 𝑣 ∈ ( Vtx ‘ 𝑔 ) ( ( VtxDeg ‘ 𝑔 ) ‘ 𝑣 ) = 𝑘 ) }
4	3	bropaex12	⊢ ( 𝐺 RegGraph 𝐾 → ( 𝐺 ∈ V ∧ 𝐾 ∈ V ) )
5	1 2	isrgr	⊢ ( ( 𝐺 ∈ V ∧ 𝐾 ∈ V ) → ( 𝐺 RegGraph 𝐾 ↔ ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
6	5	biimpd	⊢ ( ( 𝐺 ∈ V ∧ 𝐾 ∈ V ) → ( 𝐺 RegGraph 𝐾 → ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
7	4 6	mpcom	⊢ ( 𝐺 RegGraph 𝐾 → ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) )