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Definition df-mat 18693
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 18692 . 2  class Mat
2 vn . . 3  setvar  n
3 vr . . 3  setvar  r
4 cfn 7516 . . 3  class  Fin
5 cvv 3113 . . 3  class  _V
63cv 1378 . . . . 5  class  r
72cv 1378 . . . . . 6  class  n
87, 7cxp 4997 . . . . 5  class  ( n  X.  n )
9 cfrlm 18560 . . . . 5  class freeLMod
106, 8, 9co 6283 . . . 4  class  ( r freeLMod  ( n  X.  n
) )
11 cnx 14486 . . . . . 6  class  ndx
12 cmulr 14555 . . . . . 6  class  .r
1311, 12cfv 5587 . . . . 5  class  ( .r
`  ndx )
147, 7, 7cotp 4035 . . . . . 6  class  <. n ,  n ,  n >.
15 cmmul 18668 . . . . . 6  class maMul
166, 14, 15co 6283 . . . . 5  class  ( r maMul  <. n ,  n ,  n >. )
1713, 16cop 4033 . . . 4  class  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. )
>.
18 csts 14487 . . . 4  class sSet
1910, 17, 18co 6283 . . 3  class  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. )
202, 3, 4, 5, 19cmpt2 6285 . 2  class  ( n  e.  Fin ,  r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. ) )
211, 20wceq 1379 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  18694  matbas0  18695  matval  18696  matrcl  18697  mdetfval  18871  madufval  18922
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