Metamath Proof Explorer


Theorem dmatbas

Description: The set of all N x N diagonal matrices over (the ring) R is the base set of the algebra of N x N diagonal matrices over (the ring) R . (Contributed by AV, 8-Dec-2019)

Ref Expression
Hypotheses dmatbas.a 𝐴 = ( 𝑁 Mat 𝑅 )
dmatbas.b 𝐵 = ( Base ‘ 𝐴 )
dmatbas.0 0 = ( 0g𝑅 )
dmatbas.d 𝐷 = ( 𝑁 DMat 𝑅 )
Assertion dmatbas ( ( 𝑁 ∈ Fin ∧ 𝑅𝑉 ) → 𝐷 = ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 dmatbas.a 𝐴 = ( 𝑁 Mat 𝑅 )
2 dmatbas.b 𝐵 = ( Base ‘ 𝐴 )
3 dmatbas.0 0 = ( 0g𝑅 )
4 dmatbas.d 𝐷 = ( 𝑁 DMat 𝑅 )
5 1 2 3 4 dmatval ( ( 𝑁 ∈ Fin ∧ 𝑅𝑉 ) → 𝐷 = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
6 elex ( 𝑅𝑉𝑅 ∈ V )
7 eqid ( 𝑁 DMatALT 𝑅 ) = ( 𝑁 DMatALT 𝑅 )
8 1 2 3 7 dmatALTbas ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
9 6 8 sylan2 ( ( 𝑁 ∈ Fin ∧ 𝑅𝑉 ) → ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
10 5 9 eqtr4d ( ( 𝑁 ∈ Fin ∧ 𝑅𝑉 ) → 𝐷 = ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) )