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Theorem scmatel 18774
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 18773 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  S  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
87eleq2d 2537 . 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 2471 . . . 4  |-  ( m  =  M  ->  (
m  =  ( c 
.x.  .1.  )  <->  M  =  ( c  .x.  .1.  ) ) )
109rexbidv 2973 . . 3  |-  ( m  =  M  ->  ( E. c  e.  K  m  =  ( c  .x.  .1.  )  <->  E. c  e.  K  M  =  ( c  .x.  .1.  ) ) )
1110elrab 3261 . 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 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   ` cfv 5586  (class class class)co 6282   Fincfn 7513   Basecbs 14486   .scvsca 14555   1rcur 16943   Mat cmat 18676   ScMat cscmat 18758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-scmat 18760
This theorem is referenced by:  scmatscmid  18775  scmatmat  18778  scmatid  18783  scmataddcl  18785  scmatsubcl  18786  scmatmulcl  18787  smatvscl  18793  scmatrhmcl  18797  mat0scmat  18807  mat1scmat  18808  chmaidscmat  19116
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