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

Theorem marepvfval 19521
Description: First 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
marepvfval  |-  Q  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) )
Distinct variable groups:    B, m, v    i, N, j, k, m, v    R, i, j, k, m, v   
m, V, v
Allowed substitution hints:    A( v, i, j, k, m)    B( i, j, k)    Q( v, i, j, k, m)    V( i, j, k)

Proof of Theorem marepvfval
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvfval.q . 2  |-  Q  =  ( N matRepV  R )
2 marepvfval.b . . . . . 6  |-  B  =  ( Base `  A
)
3 fvex 5891 . . . . . 6  |-  ( Base `  A )  e.  _V
42, 3eqeltri 2513 . . . . 5  |-  B  e. 
_V
5 marepvfval.v . . . . . . 7  |-  V  =  ( ( Base `  R
)  ^m  N )
6 ovex 6333 . . . . . . 7  |-  ( (
Base `  R )  ^m  N )  e.  _V
75, 6eqeltri 2513 . . . . . 6  |-  V  e. 
_V
87a1i 11 . . . . 5  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  V  e.  _V )
9 mpt2exga 6883 . . . . 5  |-  ( ( B  e.  _V  /\  V  e.  _V )  ->  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) )  e.  _V )
104, 8, 9sylancr 667 . . . 4  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) )  e.  _V )
11 oveq12 6314 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
12 marepvfval.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
1311, 12syl6eqr 2488 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
1413fveq2d 5885 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
1514, 2syl6eqr 2488 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
16 fveq2 5881 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
1716adantl 467 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  r
)  =  ( Base `  R ) )
18 simpl 458 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
1917, 18oveq12d 6323 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( Base `  r
)  ^m  n )  =  ( ( Base `  R )  ^m  N
) )
2019, 5syl6eqr 2488 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( Base `  r
)  ^m  n )  =  V )
21 eqidd 2430 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) )  =  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) )
2218, 18, 21mpt2eq123dv 6367 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( i  e.  n ,  j  e.  n  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) )
2318, 22mpteq12dv 4504 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( k  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) )
2415, 20, 23mpt2eq123dv 6367 . . . . 5  |-  ( ( n  =  N  /\  r  =  R )  ->  ( m  e.  (
Base `  ( n Mat  r ) ) ,  v  e.  ( (
Base `  r )  ^m  n )  |->  ( k  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( j  =  k ,  ( v `  i
) ,  ( i m j ) ) ) ) )  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) ) )
25 df-marepv 19515 . . . . 5  |- matRepV  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) ) ,  v  e.  ( ( Base `  r
)  ^m  n )  |->  ( k  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) ) )
2624, 25ovmpt2ga 6440 . . . 4  |-  ( ( N  e.  _V  /\  R  e.  _V  /\  (
m  e.  B , 
v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) )  e.  _V )  -> 
( N matRepV  R )  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) ) )
2710, 26mpd3an3 1361 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRepV  R )  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) ) )
2825mpt2ndm0 6524 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRepV  R )  =  (/) )
29 mpt20 6375 . . . . 5  |-  ( m  e.  (/) ,  v  e.  ( ( Base `  R
)  ^m  N )  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) )  =  (/)
3028, 29syl6eqr 2488 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRepV  R )  =  ( m  e.  (/) ,  v  e.  ( ( Base `  R
)  ^m  N )  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) ) )
3112fveq2i 5884 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
322, 31eqtri 2458 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
33 matbas0pc 19365 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
3432, 33syl5eq 2482 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
35 mpt2eq12 6365 . . . . 5  |-  ( ( B  =  (/)  /\  V  =  ( ( Base `  R )  ^m  N
) )  ->  (
m  e.  B , 
v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) )  =  ( m  e.  (/) ,  v  e.  ( ( Base `  R
)  ^m  N )  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) ) )
3634, 5, 35sylancl 666 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) )  =  ( m  e.  (/) ,  v  e.  ( ( Base `  R
)  ^m  N )  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) ) )
3730, 36eqtr4d 2473 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRepV  R )  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) ) )
3827, 37pm2.61i 167 . 2  |-  ( N matRepV  R )  =  ( m  e.  B , 
v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) ) ) )
391, 38eqtri 2458 1  |-  Q  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   ifcif 3915    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307    ^m cmap 7480   Basecbs 15084   Mat cmat 19363   matRepV cmatrepV 19513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-slot 15088  df-base 15089  df-mat 19364  df-marepv 19515
This theorem is referenced by:  marepvval0  19522
  Copyright terms: Public domain W3C validator