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Theorem dmatcomp 18414
Description: The components of diagonal matrices. (Contributed by AV, 22-Jul-2019.)
Hypotheses
Ref Expression
mamucl.b  |-  B  =  ( Base `  R
)
mamucl.r  |-  ( ph  ->  R  e.  Ring )
mamudiag.o  |-  .1.  =  ( 1r `  R )
mamudiag.z  |-  .0.  =  ( 0g `  R )
mamudiag.i  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
mamudiag.m  |-  ( ph  ->  M  e.  Fin )
Assertion
Ref Expression
dmatcomp  |-  ( ( A  e.  M  /\  J  e.  M )  ->  ( A I J )  =  if ( A  =  J ,  .1.  ,  .0.  ) )
Distinct variable groups:    i, j, B    i, M, j    ph, i,
j    .1. , i, j    .0. , i, j    A, i, j   
i, J, j
Allowed substitution hints:    R( i, j)    I( i, j)

Proof of Theorem dmatcomp
StepHypRef Expression
1 eqeq1 2455 . . 3  |-  ( i  =  A  ->  (
i  =  j  <->  A  =  j ) )
21ifbid 3911 . 2  |-  ( i  =  A  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if ( A  =  j ,  .1.  ,  .0.  ) )
3 eqeq2 2466 . . 3  |-  ( j  =  J  ->  ( A  =  j  <->  A  =  J ) )
43ifbid 3911 . 2  |-  ( j  =  J  ->  if ( A  =  j ,  .1.  ,  .0.  )  =  if ( A  =  J ,  .1.  ,  .0.  ) )
5 mamudiag.i . 2  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
6 mamudiag.o . . . 4  |-  .1.  =  ( 1r `  R )
7 fvex 5801 . . . 4  |-  ( 1r
`  R )  e. 
_V
86, 7eqeltri 2535 . . 3  |-  .1.  e.  _V
9 mamudiag.z . . . 4  |-  .0.  =  ( 0g `  R )
10 fvex 5801 . . . 4  |-  ( 0g
`  R )  e. 
_V
119, 10eqeltri 2535 . . 3  |-  .0.  e.  _V
128, 11ifex 3958 . 2  |-  if ( A  =  J ,  .1.  ,  .0.  )  e. 
_V
132, 4, 5, 12ovmpt2 6328 1  |-  ( ( A  e.  M  /\  J  e.  M )  ->  ( A I J )  =  if ( A  =  J ,  .1.  ,  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070   ifcif 3891   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   Fincfn 7412   Basecbs 14278   0gc0g 14482   1rcur 16710   Ringcrg 16753
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197
This theorem is referenced by:  mamulid  18415  mamurid  18416
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