MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  marepvval0 Structured version   Unicode version

Theorem marepvval0 18499
Description: Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
Hypotheses
Ref Expression
marepvfval.a  |-  A  =  ( N Mat  R )
marepvfval.b  |-  B  =  ( Base `  A
)
marepvfval.q  |-  Q  =  ( N matRepV  R )
marepvfval.v  |-  V  =  ( ( Base `  R
)  ^m  N )
Assertion
Ref Expression
marepvval0  |-  ( ( M  e.  B  /\  C  e.  V )  ->  ( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) ) )
Distinct variable groups:    i, N, j, k    R, i, j, k    C, i, j, k   
i, M, j, k
Allowed substitution hints:    A( i, j, k)    B( i, j, k)    Q( i, j, k)    V( i, j, k)

Proof of Theorem marepvval0
Dummy variables  m  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.a . . . . . 6  |-  A  =  ( N Mat  R )
2 marepvfval.b . . . . . 6  |-  B  =  ( Base `  A
)
31, 2matrcl 18432 . . . . 5  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . . 4  |-  ( M  e.  B  ->  N  e.  Fin )
54adantr 465 . . 3  |-  ( ( M  e.  B  /\  C  e.  V )  ->  N  e.  Fin )
6 mptexg 6051 . . 3  |-  ( N  e.  Fin  ->  (
k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) ) )  e. 
_V )
75, 6syl 16 . 2  |-  ( ( M  e.  B  /\  C  e.  V )  ->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `
 i ) ,  ( i M j ) ) ) )  e.  _V )
8 fveq1 5793 . . . . . . 7  |-  ( c  =  C  ->  (
c `  i )  =  ( C `  i ) )
98adantl 466 . . . . . 6  |-  ( ( m  =  M  /\  c  =  C )  ->  ( c `  i
)  =  ( C `
 i ) )
10 oveq 6201 . . . . . . 7  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
1110adantr 465 . . . . . 6  |-  ( ( m  =  M  /\  c  =  C )  ->  ( i m j )  =  ( i M j ) )
129, 11ifeq12d 3912 . . . . 5  |-  ( ( m  =  M  /\  c  =  C )  ->  if ( j  =  k ,  ( c `
 i ) ,  ( i m j ) )  =  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) )
1312mpt2eq3dv 6256 . . . 4  |-  ( ( m  =  M  /\  c  =  C )  ->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( c `
 i ) ,  ( i m j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `
 i ) ,  ( i M j ) ) ) )
1413mpteq2dv 4482 . . 3  |-  ( ( m  =  M  /\  c  =  C )  ->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( c `
 i ) ,  ( i m j ) ) ) )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  ( i M j ) ) ) ) )
15 marepvfval.q . . . 4  |-  Q  =  ( N matRepV  R )
16 marepvfval.v . . . 4  |-  V  =  ( ( Base `  R
)  ^m  N )
171, 2, 15, 16marepvfval 18498 . . 3  |-  Q  =  ( m  e.  B ,  c  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( c `
 i ) ,  ( i m j ) ) ) ) )
1814, 17ovmpt2ga 6325 . 2  |-  ( ( M  e.  B  /\  C  e.  V  /\  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `
 i ) ,  ( i M j ) ) ) )  e.  _V )  -> 
( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) ) )
197, 18mpd3an3 1316 1  |-  ( ( M  e.  B  /\  C  e.  V )  ->  ( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i
) ,  ( i M j ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3072   ifcif 3894    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197    ^m cmap 7319   Fincfn 7415   Basecbs 14287   Mat cmat 18400   matRepV cmatrepV 18490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-slot 14291  df-base 14292  df-mat 18402  df-marepv 18492
This theorem is referenced by:  marepvval  18500
  Copyright terms: Public domain W3C validator