Theorem dmatALTbas

Description: The base set of the algebra of N x N diagonal matrices over a ring R , i.e. the set of all N x N diagonal matrices 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	dmatALTbas	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( 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	dmatALTval	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → 𝐷 = ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) )
6	5	fveq2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = ( Base ‘ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) )
7	2	fvexi	⊢ 𝐵 ∈ V
8		rabexg	⊢ ( 𝐵 ∈ V → { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V )
9	7 8	mp1i	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V )
10		eqid	⊢ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) = ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
11	10 2	ressbas	⊢ ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V → ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = ( Base ‘ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) )
12	9 11	syl	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = ( Base ‘ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) )
13		inrab2	⊢ ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = { 𝑚 ∈ ( 𝐵 ∩ 𝐵 ) ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) }
14		inidm	⊢ ( 𝐵 ∩ 𝐵 ) = 𝐵
15		rabeq	⊢ ( ( 𝐵 ∩ 𝐵 ) = 𝐵 → { 𝑚 ∈ ( 𝐵 ∩ 𝐵 ) ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
16	14 15	mp1i	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → { 𝑚 ∈ ( 𝐵 ∩ 𝐵 ) ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
17	13 16	eqtrid	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
18	6 12 17	3eqtr2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )