Theorem dmatmat

Description: An N x N diagonal matrix over (the ring) R is an N x N matrix over (the ring) R . (Contributed by AV, 18-Dec-2019)

	Ref	Expression
Hypotheses	dmatval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	dmatval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	dmatval.0	⊢ 0 = ( 0g ‘ 𝑅 )
	dmatval.d	⊢ 𝐷 = ( 𝑁 DMat 𝑅 )
Assertion	dmatmat	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dmatval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2		dmatval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
3		dmatval.0	⊢ 0 = ( 0g ‘ 𝑅 )
4		dmatval.d	⊢ 𝐷 = ( 𝑁 DMat 𝑅 )
5	1 2 3 4	dmatel	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) )
6		simpl	⊢ ( ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) → 𝑀 ∈ 𝐵 )
7	5 6	biimtrdi	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵 ) )