Theorem dmatALTbasel

Description: An element of the base set of the algebra of N x N diagonal matrices over a ring R , i.e. an N x N diagonal matrix over the ring R . (Contributed by AV, 8-Dec-2019)

	Ref	Expression
Hypotheses	dmatALTval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	dmatALTval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	dmatALTval.0	⊢ 0 = ( 0g ‘ 𝑅 )
	dmatALTval.d	⊢ 𝐷 = ( 𝑁 DMatALT 𝑅 )
Assertion	dmatALTbasel	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( 𝑀 ∈ ( Base ‘ 𝐷 ) ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dmatALTval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2		dmatALTval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
3		dmatALTval.0	⊢ 0 = ( 0g ‘ 𝑅 )
4		dmatALTval.d	⊢ 𝐷 = ( 𝑁 DMatALT 𝑅 )
5	1 2 3 4	dmatALTbas	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
6	5	eleq2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( 𝑀 ∈ ( Base ‘ 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 ∈ ( Base ‘ 𝐷 ) ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) )