Theorem isrusgr0

Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018) (Revised by AV, 26-Dec-2020)

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

Proof

Step	Hyp	Ref	Expression
1		isrusgr0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isrusgr0.d	⊢ 𝐷 = ( VtxDeg ‘ 𝐺 )
3		isrusgr	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( 𝐺 RegUSGraph 𝐾 ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) ) )
4	1 2	isrgr	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( 𝐺 RegGraph 𝐾 ↔ ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
5	4	anbi2d	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) ↔ ( 𝐺 ∈ USGraph ∧ ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) ) )
6		3anass	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ↔ ( 𝐺 ∈ USGraph ∧ ( 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
7	5 6	bitr4di	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) ↔ ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )
8	3 7	bitrd	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( 𝐺 RegUSGraph 𝐾 ↔ ( 𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐷 ‘ 𝑣 ) = 𝐾 ) ) )