Metamath Proof Explorer


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 ˙ = 0 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 ˙ = 0 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 ˙