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Definition df-mat 19482
Description: Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Assertion
Ref Expression
df-mat  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
Distinct variable group:    n, r

Detailed syntax breakdown of Definition df-mat
StepHypRef Expression
1 cmat 19481 . 2  class Mat
2 vn . . 3  setvar  n
3 vr . . 3  setvar  r
4 cfn 7595 . . 3  class  Fin
5 cvv 3057 . . 3  class  _V
63cv 1454 . . . . 5  class  r
72cv 1454 . . . . . 6  class  n
87, 7cxp 4851 . . . . 5  class  ( n  X.  n )
9 cfrlm 19358 . . . . 5  class freeLMod
106, 8, 9co 6315 . . . 4  class  ( r freeLMod  ( n  X.  n
) )
11 cnx 15167 . . . . . 6  class  ndx
12 cmulr 15240 . . . . . 6  class  .r
1311, 12cfv 5601 . . . . 5  class  ( .r
`  ndx )
147, 7, 7cotp 3988 . . . . . 6  class  <. n ,  n ,  n >.
15 cmmul 19457 . . . . . 6  class maMul
166, 14, 15co 6315 . . . . 5  class  ( r maMul  <. n ,  n ,  n >. )
1713, 16cop 3986 . . . 4  class  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. )
>.
18 csts 15168 . . . 4  class sSet
1910, 17, 18co 6315 . . 3  class  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. )
202, 3, 4, 5, 19cmpt2 6317 . 2  class  ( n  e.  Fin ,  r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. ) )
211, 20wceq 1455 1  wff Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
Colors of variables: wff setvar class
This definition is referenced by:  matbas0pc  19483  matbas0  19484  matval  19485  matrcl  19486  mdetfval  19660  madufval  19711
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