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Theorem minmar1eval 18588
Description: An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
minmar1fval.a  |-  A  =  ( N Mat  R )
minmar1fval.b  |-  B  =  ( Base `  A
)
minmar1fval.q  |-  Q  =  ( N minMatR1  R )
minmar1fval.o  |-  .1.  =  ( 1r `  R )
minmar1fval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
minmar1eval  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( K ( Q `  M
) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  .1.  ,  .0.  ) ,  ( I M J ) ) )

Proof of Theorem minmar1eval
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.a . . . . 5  |-  A  =  ( N Mat  R )
2 minmar1fval.b . . . . 5  |-  B  =  ( Base `  A
)
3 minmar1fval.q . . . . 5  |-  Q  =  ( N minMatR1  R )
4 minmar1fval.o . . . . 5  |-  .1.  =  ( 1r `  R )
5 minmar1fval.z . . . . 5  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5minmar1val 18587 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( K ( Q `
 M ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
763expb 1189 . . 3  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
) )  ->  ( K ( Q `  M ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
873adant3 1008 . 2  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( K ( Q `
 M ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
9 simp3l 1016 . . 3  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  ->  I  e.  N )
10 simpl3r 1044 . . 3  |-  ( ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  /\  i  =  I )  ->  J  e.  N )
11 fvex 5810 . . . . . . 7  |-  ( 1r
`  R )  e. 
_V
124, 11eqeltri 2538 . . . . . 6  |-  .1.  e.  _V
13 fvex 5810 . . . . . . 7  |-  ( 0g
`  R )  e. 
_V
145, 13eqeltri 2538 . . . . . 6  |-  .0.  e.  _V
1512, 14ifex 3967 . . . . 5  |-  if ( j  =  L ,  .1.  ,  .0.  )  e. 
_V
16 ovex 6226 . . . . 5  |-  ( i M j )  e. 
_V
1715, 16ifex 3967 . . . 4  |-  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) )  e.  _V
1817a1i 11 . . 3  |-  ( ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  /\  ( i  =  I  /\  j  =  J ) )  ->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) )  e.  _V )
19 eqeq1 2458 . . . . . 6  |-  ( i  =  I  ->  (
i  =  K  <->  I  =  K ) )
2019adantr 465 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i  =  K  <-> 
I  =  K ) )
21 eqeq1 2458 . . . . . . 7  |-  ( j  =  J  ->  (
j  =  L  <->  J  =  L ) )
2221adantl 466 . . . . . 6  |-  ( ( i  =  I  /\  j  =  J )  ->  ( j  =  L  <-> 
J  =  L ) )
2322ifbid 3920 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( j  =  L ,  .1.  ,  .0.  )  =  if ( J  =  L ,  .1.  ,  .0.  )
)
24 oveq12 6210 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i M j )  =  ( I M J ) )
2520, 23, 24ifbieq12d 3925 . . . 4  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) )  =  if ( I  =  K ,  if ( J  =  L ,  .1.  ,  .0.  ) ,  ( I M J ) ) )
2625adantl 466 . . 3  |-  ( ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  /\  ( i  =  I  /\  j  =  J ) )  ->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) )  =  if ( I  =  K ,  if ( J  =  L ,  .1.  ,  .0.  ) ,  ( I M J ) ) )
279, 10, 18, 26ovmpt2dv2 6335 . 2  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( ( K ( Q `  M ) L )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  ->  (
I ( K ( Q `  M ) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  .1.  ,  .0.  ) ,  ( I M J ) ) ) )
288, 27mpd 15 1  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( K ( Q `  M
) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  .1.  ,  .0.  ) ,  ( I M J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   ifcif 3900   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   Basecbs 14293   0gc0g 14498   1rcur 16726   Mat cmat 18406   minMatR1 cminmar1 18572
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-slot 14297  df-base 14298  df-mat 18408  df-minmar1 18574
This theorem is referenced by: (None)
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