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Theorem marepvfval 18388
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 5713 . . . . . 6  |-  ( Base `  A )  e.  _V
42, 3eqeltri 2513 . . . . 5  |-  B  e. 
_V
5 marepvfval.v . . . . . . 7  |-  V  =  ( ( Base `  R
)  ^m  N )
6 ovex 6128 . . . . . . 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 6661 . . . . 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 6112 . . . . . . . . 9  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  ( N Mat  R
) )
12 marepvfval.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
1311, 12syl6eqr 2493 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  ( n Mat  r )  =  A )
1413fveq2d 5707 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  ( Base `  A
) )
1514, 2syl6eqr 2493 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( Base `  (
n Mat  r ) )  =  B )
16 fveq2 5703 . . . . . . . . 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 6121 . . . . . . 7  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( Base `  r
)  ^m  n )  =  ( ( Base `  R )  ^m  N
) )
2019, 5syl6eqr 2493 . . . . . 6  |-  ( ( n  =  N  /\  r  =  R )  ->  ( ( Base `  r
)  ^m  n )  =  V )
21 eqidd 2444 . . . . . . . 8  |-  ( ( n  =  N  /\  r  =  R )  ->  if ( j  =  k ,  ( v `
 i ) ,  ( i m j ) )  =  if ( j  =  k ,  ( v `  i ) ,  ( i m j ) ) )
2218, 18, 21mpt2eq123dv 6160 . . . . . . 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 4382 . . . . . 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 6160 . . . . 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 18382 . . . . 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 6232 . . . 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 1315 . . 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 6751 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N matRepV  R )  =  (/) )
29 mpt20 6168 . . . . 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 2493 . . . 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 5706 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
322, 31eqtri 2463 . . . . . 6  |-  B  =  ( Base `  ( N Mat  R ) )
33 matbas0pc 18298 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
3432, 33syl5eq 2487 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
35 mpt2eq12 6158 . . . . 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 2478 . . 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 2463 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 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649   ifcif 3803    e. cmpt 4362   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105    ^m cmap 7226   Basecbs 14186   Mat cmat 18292   matRepV cmatrepV 18380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-slot 14190  df-base 14191  df-mat 18294  df-marepv 18382
This theorem is referenced by:  marepvval0  18389
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