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Theorem minmar1eval 19236
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 19235 . . . 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 1195 . . 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 1014 . 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 1022 . . 3  |-  ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  ->  I  e.  N )
10 simpl3r 1050 . . 3  |-  ( ( ( M  e.  B  /\  ( K  e.  N  /\  L  e.  N
)  /\  ( I  e.  N  /\  J  e.  N ) )  /\  i  =  I )  ->  J  e.  N )
11 fvex 5784 . . . . . . 7  |-  ( 1r
`  R )  e. 
_V
124, 11eqeltri 2466 . . . . . 6  |-  .1.  e.  _V
13 fvex 5784 . . . . . . 7  |-  ( 0g
`  R )  e. 
_V
145, 13eqeltri 2466 . . . . . 6  |-  .0.  e.  _V
1512, 14ifex 3925 . . . . 5  |-  if ( j  =  L ,  .1.  ,  .0.  )  e. 
_V
16 ovex 6224 . . . . 5  |-  ( i M j )  e. 
_V
1715, 16ifex 3925 . . . 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 2386 . . . . . 6  |-  ( i  =  I  ->  (
i  =  K  <->  I  =  K ) )
2019adantr 463 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i  =  K  <-> 
I  =  K ) )
21 eqeq1 2386 . . . . . . 7  |-  ( j  =  J  ->  (
j  =  L  <->  J  =  L ) )
2221adantl 464 . . . . . 6  |-  ( ( i  =  I  /\  j  =  J )  ->  ( j  =  L  <-> 
J  =  L ) )
2322ifbid 3879 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  if ( j  =  L ,  .1.  ,  .0.  )  =  if ( J  =  L ,  .1.  ,  .0.  )
)
24 oveq12 6205 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i M j )  =  ( I M J ) )
2520, 23, 24ifbieq12d 3884 . . . 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 464 . . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   _Vcvv 3034   ifcif 3857   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   Basecbs 14634   0gc0g 14847   1rcur 17266   Mat cmat 18994   minMatR1 cminmar1 19220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-slot 14638  df-base 14639  df-mat 18995  df-minmar1 19222
This theorem is referenced by: (None)
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