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Theorem scmatval 19173
Description: The set of  N x  N scalar matrices over (a ring)  R. (Contributed by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
scmatval.k  |-  K  =  ( Base `  R
)
scmatval.a  |-  A  =  ( N Mat  R )
scmatval.b  |-  B  =  ( Base `  A
)
scmatval.1  |-  .1.  =  ( 1r `  A )
scmatval.t  |-  .x.  =  ( .s `  A )
scmatval.s  |-  S  =  ( N ScMat  R )
Assertion
Ref Expression
scmatval  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  S  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
Distinct variable groups:    B, m    K, c    N, c, m    R, c, m
Allowed substitution hints:    A( m, c)    B( c)    S( m, c)    .x. ( m, c)    .1. ( m, c)    K( m)    V( m, c)

Proof of Theorem scmatval
Dummy variables  n  r  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scmatval.s . 2  |-  S  =  ( N ScMat  R )
2 df-scmat 19160 . . . 4  |- ScMat  =  ( n  e.  Fin , 
r  e.  _V  |->  [_ ( n Mat  r )  /  a ]_ {
m  e.  ( Base `  a )  |  E. c  e.  ( Base `  r ) m  =  ( c ( .s
`  a ) ( 1r `  a ) ) } )
32a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  -> ScMat  =  ( n  e. 
Fin ,  r  e.  _V  |->  [_ ( n Mat  r
)  /  a ]_ { m  e.  ( Base `  a )  |  E. c  e.  (
Base `  r )
m  =  ( c ( .s `  a
) ( 1r `  a ) ) } ) )
4 ovex 6298 . . . . . 6  |-  ( n Mat  r )  e.  _V
54a1i 11 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  V )  /\  ( n  =  N  /\  r  =  R ) )  -> 
( n Mat  r )  e.  _V )
6 fveq2 5848 . . . . . . 7  |-  ( a  =  ( n Mat  r
)  ->  ( Base `  a )  =  (
Base `  ( n Mat  r ) ) )
7 fveq2 5848 . . . . . . . . . 10  |-  ( a  =  ( n Mat  r
)  ->  ( .s `  a )  =  ( .s `  ( n Mat  r ) ) )
8 eqidd 2455 . . . . . . . . . 10  |-  ( a  =  ( n Mat  r
)  ->  c  =  c )
9 fveq2 5848 . . . . . . . . . 10  |-  ( a  =  ( n Mat  r
)  ->  ( 1r `  a )  =  ( 1r `  ( n Mat  r ) ) )
107, 8, 9oveq123d 6291 . . . . . . . . 9  |-  ( a  =  ( n Mat  r
)  ->  ( c
( .s `  a
) ( 1r `  a ) )  =  ( c ( .s
`  ( n Mat  r
) ) ( 1r
`  ( n Mat  r
) ) ) )
1110eqeq2d 2468 . . . . . . . 8  |-  ( a  =  ( n Mat  r
)  ->  ( m  =  ( c ( .s `  a ) ( 1r `  a
) )  <->  m  =  ( c ( .s
`  ( n Mat  r
) ) ( 1r
`  ( n Mat  r
) ) ) ) )
1211rexbidv 2965 . . . . . . 7  |-  ( a  =  ( n Mat  r
)  ->  ( E. c  e.  ( Base `  r ) m  =  ( c ( .s
`  a ) ( 1r `  a ) )  <->  E. c  e.  (
Base `  r )
m  =  ( c ( .s `  (
n Mat  r ) ) ( 1r `  (
n Mat  r ) ) ) ) )
136, 12rabeqbidv 3101 . . . . . 6  |-  ( a  =  ( n Mat  r
)  ->  { m  e.  ( Base `  a
)  |  E. c  e.  ( Base `  r
) m  =  ( c ( .s `  a ) ( 1r
`  a ) ) }  =  { m  e.  ( Base `  (
n Mat  r ) )  |  E. c  e.  ( Base `  r
) m  =  ( c ( .s `  ( n Mat  r )
) ( 1r `  ( n Mat  r )
) ) } )
1413adantl 464 . . . . 5  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  V )  /\  (
n  =  N  /\  r  =  R )
)  /\  a  =  ( n Mat  r )
)  ->  { m  e.  ( Base `  a
)  |  E. c  e.  ( Base `  r
) m  =  ( c ( .s `  a ) ( 1r
`  a ) ) }  =  { m  e.  ( Base `  (
n Mat  r ) )  |  E. c  e.  ( Base `  r
) m  =  ( c ( .s `  ( n Mat  r )
) ( 1r `  ( n Mat  r )
) ) } )
155, 14csbied 3447 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  V )  /\  ( n  =  N  /\  r  =  R ) )  ->  [_ ( n Mat  r )  /  a ]_ {
m  e.  ( Base `  a )  |  E. c  e.  ( Base `  r ) m  =  ( c ( .s
`  a ) ( 1r `  a ) ) }  =  {
m  e.  ( Base `  ( n Mat  r ) )  |  E. c  e.  ( Base `  r
) m  =  ( c ( .s `  ( n Mat  r )
) ( 1r `  ( n Mat  r )
) ) } )
16 oveq12 6279 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
1716fveq2d 5852 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  ( N Mat  R ) ) )
18 scmatval.b . . . . . . . 8  |-  B  =  ( Base `  A
)
19 scmatval.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
2019fveq2i 5851 . . . . . . . 8  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
2118, 20eqtri 2483 . . . . . . 7  |-  B  =  ( Base `  ( N Mat  R ) )
2217, 21syl6eqr 2513 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
23 fveq2 5848 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
24 scmatval.k . . . . . . . . 9  |-  K  =  ( Base `  R
)
2523, 24syl6eqr 2513 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  K )
2625adantl 464 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  r
)  =  K )
2716fveq2d 5852 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .s `  (
n Mat  r ) )  =  ( .s `  ( N Mat  R )
) )
28 scmatval.t . . . . . . . . . . 11  |-  .x.  =  ( .s `  A )
2919fveq2i 5851 . . . . . . . . . . 11  |-  ( .s
`  A )  =  ( .s `  ( N Mat  R ) )
3028, 29eqtri 2483 . . . . . . . . . 10  |-  .x.  =  ( .s `  ( N Mat 
R ) )
3127, 30syl6eqr 2513 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( .s `  (
n Mat  r ) )  =  .x.  )
32 eqidd 2455 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  c  =  c )
3316fveq2d 5852 . . . . . . . . . 10  |-  ( ( n  =  N  /\  r  =  R )  ->  ( 1r `  (
n Mat  r ) )  =  ( 1r `  ( N Mat  R )
) )
34 scmatval.1 . . . . . . . . . . 11  |-  .1.  =  ( 1r `  A )
3519fveq2i 5851 . . . . . . . . . . 11  |-  ( 1r
`  A )  =  ( 1r `  ( N Mat  R ) )
3634, 35eqtri 2483 . . . . . . . . . 10  |-  .1.  =  ( 1r `  ( N Mat 
R ) )
3733, 36syl6eqr 2513 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( 1r `  (
n Mat  r ) )  =  .1.  )
3831, 32, 37oveq123d 6291 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( c ( .s
`  ( n Mat  r
) ) ( 1r
`  ( n Mat  r
) ) )  =  ( c  .x.  .1.  ) )
3938eqeq2d 2468 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  =  ( c ( .s `  ( n Mat  r )
) ( 1r `  ( n Mat  r )
) )  <->  m  =  ( c  .x.  .1.  ) ) )
4026, 39rexeqbidv 3066 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( E. c  e.  ( Base `  r
) m  =  ( c ( .s `  ( n Mat  r )
) ( 1r `  ( n Mat  r )
) )  <->  E. c  e.  K  m  =  ( c  .x.  .1.  ) ) )
4122, 40rabeqbidv 3101 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  { m  e.  (
Base `  ( n Mat  r ) )  |  E. c  e.  (
Base `  r )
m  =  ( c ( .s `  (
n Mat  r ) ) ( 1r `  (
n Mat  r ) ) ) }  =  {
m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
4241adantl 464 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  V )  /\  ( n  =  N  /\  r  =  R ) )  ->  { m  e.  ( Base `  ( n Mat  r
) )  |  E. c  e.  ( Base `  r ) m  =  ( c ( .s
`  ( n Mat  r
) ) ( 1r
`  ( n Mat  r
) ) ) }  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
4315, 42eqtrd 2495 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  V )  /\  ( n  =  N  /\  r  =  R ) )  ->  [_ ( n Mat  r )  /  a ]_ {
m  e.  ( Base `  a )  |  E. c  e.  ( Base `  r ) m  =  ( c ( .s
`  a ) ( 1r `  a ) ) }  =  {
m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
44 simpl 455 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  N  e.  Fin )
45 elex 3115 . . . 4  |-  ( R  e.  V  ->  R  e.  _V )
4645adantl 464 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  R  e.  _V )
47 fvex 5858 . . . . . 6  |-  ( Base `  A )  e.  _V
4818, 47eqeltri 2538 . . . . 5  |-  B  e. 
_V
4948rabex 4588 . . . 4  |-  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) }  e.  _V
5049a1i 11 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) }  e.  _V )
513, 43, 44, 46, 50ovmpt2d 6403 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( N ScMat  R )  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
521, 51syl5eq 2507 1  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  S  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106   [_csb 3420   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Fincfn 7509   Basecbs 14716   .scvsca 14788   1rcur 17348   Mat cmat 19076   ScMat cscmat 19158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-scmat 19160
This theorem is referenced by:  scmatel  19174  scmatmats  19180  scmatlss  19194
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