Metamath Proof Explorer


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 = ( 0g𝑅 )
dmatALTval.d 𝐷 = ( 𝑁 DMatALT 𝑅 )
Assertion dmatALTbas ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 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 = ( 0g𝑅 )
4 dmatALTval.d 𝐷 = ( 𝑁 DMatALT 𝑅 )
5 1 2 3 4 dmatALTval ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → 𝐷 = ( 𝐴s { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) )
6 5 fveq2d ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = ( Base ‘ ( 𝐴s { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) )
7 2 fvexi 𝐵 ∈ V
8 rabexg ( 𝐵 ∈ V → { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V )
9 7 8 mp1i ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V )
10 eqid ( 𝐴s { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) = ( 𝐴s { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
11 10 2 ressbas ( { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V → ( { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = ( Base ‘ ( 𝐴s { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) )
12 9 11 syl ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = ( Base ‘ ( 𝐴s { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) )
13 inrab2 ( { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = { 𝑚 ∈ ( 𝐵𝐵 ) ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) }
14 inidm ( 𝐵𝐵 ) = 𝐵
15 rabeq ( ( 𝐵𝐵 ) = 𝐵 → { 𝑚 ∈ ( 𝐵𝐵 ) ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
16 14 15 mp1i ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → { 𝑚 ∈ ( 𝐵𝐵 ) ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
17 13 16 syl5eq ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )
18 6 12 17 3eqtr2d ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = { 𝑚𝐵 ∣ ∀ 𝑖𝑁𝑗𝑁 ( 𝑖𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } )