Theorem dmatel

Description: A N x N diagonal matrix over (a 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	dmatel	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) )

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	dmatval	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐷 = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
6	5	eleq2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 ↔ 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) )
7		oveq	⊢ ( 𝑚 = 𝑀 → ( 𝑖 𝑚 𝑗 ) = ( 𝑖 𝑀 𝑗 ) )
8	7	eqeq1d	⊢ ( 𝑚 = 𝑀 → ( ( 𝑖 𝑚 𝑗 ) = 0 ↔ ( 𝑖 𝑀 𝑗 ) = 0 ) )
9	8	imbi2d	⊢ ( 𝑚 = 𝑀 → ( ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) ↔ ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) )
10	9	2ralbidv	⊢ ( 𝑚 = 𝑀 → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) )
11	10	elrab	⊢ ( 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) )
12	6 11	bitrdi	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) )