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Theorem marepvfval 18874
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 5876 . . . . . 6  |-  ( Base `  A )  e.  _V
42, 3eqeltri 2551 . . . . 5  |-  B  e. 
_V
5 marepvfval.v . . . . . . 7  |-  V  =  ( ( Base `  R
)  ^m  N )
6 ovex 6310 . . . . . . 7  |-  ( (
Base `  R )  ^m  N )  e.  _V
75, 6eqeltri 2551 . . . . . 6  |-  V  e. 
_V
87a1i 11 . . . . 5  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  V  e.  _V )
9 mpt2exga 6860 . . . . 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 663 . . . 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 6294 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
12 marepvfval.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
1311, 12syl6eqr 2526 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
1413fveq2d 5870 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
1514, 2syl6eqr 2526 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
16 fveq2 5866 . . . . . . . . 9  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
1716adantl 466 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  r
)  =  ( Base `  R ) )
18 simpl 457 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  n  =  N )
1917, 18oveq12d 6303 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( Base `  r
)  ^m  n )  =  ( ( Base `  R )  ^m  N
) )
2019, 5syl6eqr 2526 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( Base `  r
)  ^m  n )  =  V )
21 eqidd 2468 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) )  =  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) )
2218, 18, 21mpt2eq123dv 6344 . . . . . . 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 4525 . . . . . 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 6344 . . . . 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 18868 . . . . 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 6417 . . . 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 1325 . . 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 6501 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRepV  R )  =  (/) )
29 mpt20 6352 . . . . 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 2526 . . . 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 5869 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
322, 31eqtri 2496 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
33 matbas0pc 18718 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
3432, 33syl5eq 2520 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
35 mpt2eq12 6342 . . . . 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 662 . . . 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 2511 . . 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 164 . 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 2496 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ifcif 3939    |-> cmpt 4505   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287    ^m cmap 7421   Basecbs 14493   Mat cmat 18716   matRepV cmatrepV 18866
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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-reu 2821  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-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-slot 14497  df-base 14498  df-mat 18717  df-marepv 18868
This theorem is referenced by:  marepvval0  18875
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