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Theorem scmatel 19297
Description: An  N x  N scalar matrix 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
scmatel  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( M  e.  S  <->  ( M  e.  B  /\  E. c  e.  K  M  =  ( c  .x.  .1.  ) ) ) )
Distinct variable groups:    K, c    N, c    R, c    M, c
Allowed substitution hints:    A( c)    B( c)    S( c)    .x. ( c)    .1. ( c)    V( c)

Proof of Theorem scmatel
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 scmatval.k . . . 4  |-  K  =  ( Base `  R
)
2 scmatval.a . . . 4  |-  A  =  ( N Mat  R )
3 scmatval.b . . . 4  |-  B  =  ( Base `  A
)
4 scmatval.1 . . . 4  |-  .1.  =  ( 1r `  A )
5 scmatval.t . . . 4  |-  .x.  =  ( .s `  A )
6 scmatval.s . . . 4  |-  S  =  ( N ScMat  R )
71, 2, 3, 4, 5, 6scmatval 19296 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  S  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
87eleq2d 2472 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( M  e.  S  <->  M  e.  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } ) )
9 eqeq1 2406 . . . 4  |-  ( m  =  M  ->  (
m  =  ( c 
.x.  .1.  )  <->  M  =  ( c  .x.  .1.  ) ) )
109rexbidv 2917 . . 3  |-  ( m  =  M  ->  ( E. c  e.  K  m  =  ( c  .x.  .1.  )  <->  E. c  e.  K  M  =  ( c  .x.  .1.  ) ) )
1110elrab 3206 . 2  |-  ( M  e.  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) }  <->  ( M  e.  B  /\  E. c  e.  K  M  =  ( c  .x.  .1.  ) ) )
128, 11syl6bb 261 1  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( M  e.  S  <->  ( M  e.  B  /\  E. c  e.  K  M  =  ( c  .x.  .1.  ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754   {crab 2757   ` cfv 5568  (class class class)co 6277   Fincfn 7553   Basecbs 14839   .scvsca 14911   1rcur 17471   Mat cmat 19199   ScMat cscmat 19281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-scmat 19283
This theorem is referenced by:  scmatscmid  19298  scmatmat  19301  scmatid  19306  scmataddcl  19308  scmatsubcl  19309  scmatmulcl  19310  smatvscl  19316  scmatrhmcl  19320  mat0scmat  19330  mat1scmat  19331  chmaidscmat  19639
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