MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  minmar1val Structured version   Unicode version

Theorem minmar1val 18910
Description: Third substitution for the definition 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
minmar1val  |-  ( ( 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 ) ) ) )
Distinct variable groups:    i, N, j    R, i, j    i, M, j    i, K, j   
i, L, j
Allowed substitution hints:    A( i, j)    B( i, j)    Q( i, j)    .1. ( i, j)    .0. ( i, j)

Proof of Theorem minmar1val
Dummy variables  k 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 minmar1fval.a . . . 4  |-  A  =  ( N Mat  R )
2 minmar1fval.b . . . 4  |-  B  =  ( Base `  A
)
3 minmar1fval.q . . . 4  |-  Q  =  ( N minMatR1  R )
4 minmar1fval.o . . . 4  |-  .1.  =  ( 1r `  R )
5 minmar1fval.z . . . 4  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5minmar1val0 18909 . . 3  |-  ( M  e.  B  ->  ( Q `  M )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ) )
763ad2ant1 1012 . 2  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( Q `  M
)  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ) )
8 simp2 992 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  K  e.  N )
9 simpl3 996 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  k  =  K )  ->  L  e.  N )
101, 2matrcl 18674 . . . . . . . 8  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
1110simpld 459 . . . . . . 7  |-  ( M  e.  B  ->  N  e.  Fin )
1211, 11jca 532 . . . . . 6  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  N  e.  Fin ) )
13123ad2ant1 1012 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( N  e.  Fin  /\  N  e.  Fin )
)
1413adantr 465 . . . 4  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( N  e.  Fin  /\  N  e.  Fin )
)
15 mpt2exga 6849 . . . 4  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  e.  _V )
1614, 15syl 16 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  e.  _V )
17 eqeq2 2475 . . . . . . 7  |-  ( k  =  K  ->  (
i  =  k  <->  i  =  K ) )
1817adantr 465 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  =  k  <-> 
i  =  K ) )
19 eqeq2 2475 . . . . . . . 8  |-  ( l  =  L  ->  (
j  =  l  <->  j  =  L ) )
2019ifbid 3954 . . . . . . 7  |-  ( l  =  L  ->  if ( j  =  l ,  .1.  ,  .0.  )  =  if (
j  =  L ,  .1.  ,  .0.  ) )
2120adantl 466 . . . . . 6  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( j  =  l ,  .1.  ,  .0.  )  =  if ( j  =  L ,  .1.  ,  .0.  ) )
2218, 21ifbieq1d 3955 . . . . 5  |-  ( ( k  =  K  /\  l  =  L )  ->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) )  =  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) )
2322mpt2eq3dv 6338 . . . 4  |-  ( ( k  =  K  /\  l  =  L )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
2423adantl 466 . . 3  |-  ( ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N
)  /\  ( k  =  K  /\  l  =  L ) )  -> 
( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )
258, 9, 16, 24ovmpt2dv2 6411 . 2  |-  ( ( M  e.  B  /\  K  e.  N  /\  L  e.  N )  ->  ( ( Q `  M )  =  ( k  e.  N , 
l  e.  N  |->  ( i  e.  N , 
j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) )  -> 
( K ( Q `
 M ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  L ,  .1.  ,  .0.  ) ,  ( i M j ) ) ) ) )
267, 25mpd 15 1  |-  ( ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106   ifcif 3932   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   Fincfn 7506   Basecbs 14479   0gc0g 14684   1rcur 16936   Mat cmat 18669   minMatR1 cminmar1 18895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-slot 14483  df-base 14484  df-mat 18670  df-minmar1 18897
This theorem is referenced by:  minmar1eval  18911  maducoevalmin1  18914  smadiadet  18932  smadiadetglem2  18934
  Copyright terms: Public domain W3C validator