Metamath Proof Explorer


Theorem matassa

Description: Existence of the matrix algebra, see also the statement in Lang p. 505: "Then Mat_n(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015)

Ref Expression
Hypothesis matassa.a A = N Mat R
Assertion matassa N Fin R CRing A AssAlg

Proof

Step Hyp Ref Expression
1 matassa.a A = N Mat R
2 eqid Base R = Base R
3 1 2 matbas2 N Fin R CRing Base R N × N = Base A
4 1 matsca2 N Fin R CRing R = Scalar A
5 eqidd N Fin R CRing Base R = Base R
6 eqidd N Fin R CRing A = A
7 eqid R maMul N N N = R maMul N N N
8 1 7 matmulr N Fin R CRing R maMul N N N = A
9 crngring R CRing R Ring
10 1 matlmod N Fin R Ring A LMod
11 9 10 sylan2 N Fin R CRing A LMod
12 1 matring N Fin R Ring A Ring
13 9 12 sylan2 N Fin R CRing A Ring
14 9 ad2antlr N Fin R CRing x Base R y Base R N × N z Base R N × N R Ring
15 simpll N Fin R CRing x Base R y Base R N × N z Base R N × N N Fin
16 eqid R = R
17 simpr1 N Fin R CRing x Base R y Base R N × N z Base R N × N x Base R
18 simpr2 N Fin R CRing x Base R y Base R N × N z Base R N × N y Base R N × N
19 simpr3 N Fin R CRing x Base R y Base R N × N z Base R N × N z Base R N × N
20 2 14 7 15 15 15 16 17 18 19 mamuvs1 N Fin R CRing x Base R y Base R N × N z Base R N × N N × N × x R f y R maMul N N N z = N × N × x R f y R maMul N N N z
21 3 adantr N Fin R CRing x Base R y Base R N × N z Base R N × N Base R N × N = Base A
22 18 21 eleqtrd N Fin R CRing x Base R y Base R N × N z Base R N × N y Base A
23 eqid Base A = Base A
24 eqid A = A
25 eqid N × N = N × N
26 1 23 2 24 16 25 matvsca2 x Base R y Base A x A y = N × N × x R f y
27 17 22 26 syl2anc N Fin R CRing x Base R y Base R N × N z Base R N × N x A y = N × N × x R f y
28 27 oveq1d N Fin R CRing x Base R y Base R N × N z Base R N × N x A y R maMul N N N z = N × N × x R f y R maMul N N N z
29 2 14 7 15 15 15 18 19 mamucl N Fin R CRing x Base R y Base R N × N z Base R N × N y R maMul N N N z Base R N × N
30 29 21 eleqtrd N Fin R CRing x Base R y Base R N × N z Base R N × N y R maMul N N N z Base A
31 1 23 2 24 16 25 matvsca2 x Base R y R maMul N N N z Base A x A y R maMul N N N z = N × N × x R f y R maMul N N N z
32 17 30 31 syl2anc N Fin R CRing x Base R y Base R N × N z Base R N × N x A y R maMul N N N z = N × N × x R f y R maMul N N N z
33 20 28 32 3eqtr4d N Fin R CRing x Base R y Base R N × N z Base R N × N x A y R maMul N N N z = x A y R maMul N N N z
34 simplr N Fin R CRing x Base R y Base R N × N z Base R N × N R CRing
35 34 2 16 7 15 15 15 18 17 19 mamuvs2 N Fin R CRing x Base R y Base R N × N z Base R N × N y R maMul N N N N × N × x R f z = N × N × x R f y R maMul N N N z
36 19 21 eleqtrd N Fin R CRing x Base R y Base R N × N z Base R N × N z Base A
37 1 23 2 24 16 25 matvsca2 x Base R z Base A x A z = N × N × x R f z
38 17 36 37 syl2anc N Fin R CRing x Base R y Base R N × N z Base R N × N x A z = N × N × x R f z
39 38 oveq2d N Fin R CRing x Base R y Base R N × N z Base R N × N y R maMul N N N x A z = y R maMul N N N N × N × x R f z
40 35 39 32 3eqtr4d N Fin R CRing x Base R y Base R N × N z Base R N × N y R maMul N N N x A z = x A y R maMul N N N z
41 3 4 5 6 8 11 13 33 40 isassad N Fin R CRing A AssAlg