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Theorem dmatelnd 30873
Description: An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.)
Hypotheses
Ref Expression
dmatid.a  |-  A  =  ( N Mat  R )
dmatid.b  |-  B  =  ( Base `  A
)
dmatid.0  |-  .0.  =  ( 0g `  R )
dmatid.d  |-  D  =  { m  e.  B  |  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  ) }
Assertion
Ref Expression
dmatelnd  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  X  e.  D )  /\  (
I  e.  N  /\  J  e.  N  /\  I  =/=  J ) )  ->  ( I X J )  =  .0.  )
Distinct variable groups:    B, m    A, i, j, m    i, N, j, m    R, i, j    .0. , m    i, I, j    i, J, j   
i, X, j, m    .0. , i, j
Allowed substitution hints:    B( i, j)    D( i, j, m)    R( m)    I( m)    J( m)

Proof of Theorem dmatelnd
StepHypRef Expression
1 dmatid.d . . . . . 6  |-  D  =  { m  e.  B  |  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  ) }
21eleq2i 2506 . . . . 5  |-  ( X  e.  D  <->  X  e.  { m  e.  B  |  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  ) } )
3 oveq 6096 . . . . . . . . . 10  |-  ( m  =  X  ->  (
i m j )  =  ( i X j ) )
43eqeq1d 2450 . . . . . . . . 9  |-  ( m  =  X  ->  (
( i m j )  =  .0.  <->  ( i X j )  =  .0.  ) )
54imbi2d 316 . . . . . . . 8  |-  ( m  =  X  ->  (
( i  =/=  j  ->  ( i m j )  =  .0.  )  <->  ( i  =/=  j  -> 
( i X j )  =  .0.  )
) )
65ralbidv 2734 . . . . . . 7  |-  ( m  =  X  ->  ( A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  )  <->  A. j  e.  N  ( i  =/=  j  -> 
( i X j )  =  .0.  )
) )
76ralbidv 2734 . . . . . 6  |-  ( m  =  X  ->  ( A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  )  <->  A. i  e.  N  A. j  e.  N  (
i  =/=  j  -> 
( i X j )  =  .0.  )
) )
87elrab 3116 . . . . 5  |-  ( X  e.  { m  e.  B  |  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  ) }  <->  ( X  e.  B  /\  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  ) ) )
92, 8bitri 249 . . . 4  |-  ( X  e.  D  <->  ( X  e.  B  /\  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  ) ) )
10 neeq1 2615 . . . . . . . . . . 11  |-  ( i  =  I  ->  (
i  =/=  j  <->  I  =/=  j ) )
11 oveq1 6097 . . . . . . . . . . . 12  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
1211eqeq1d 2450 . . . . . . . . . . 11  |-  ( i  =  I  ->  (
( i X j )  =  .0.  <->  ( I X j )  =  .0.  ) )
1310, 12imbi12d 320 . . . . . . . . . 10  |-  ( i  =  I  ->  (
( i  =/=  j  ->  ( i X j )  =  .0.  )  <->  ( I  =/=  j  -> 
( I X j )  =  .0.  )
) )
14 neeq2 2616 . . . . . . . . . . 11  |-  ( j  =  J  ->  (
I  =/=  j  <->  I  =/=  J ) )
15 oveq2 6098 . . . . . . . . . . . 12  |-  ( j  =  J  ->  (
I X j )  =  ( I X J ) )
1615eqeq1d 2450 . . . . . . . . . . 11  |-  ( j  =  J  ->  (
( I X j )  =  .0.  <->  ( I X J )  =  .0.  ) )
1714, 16imbi12d 320 . . . . . . . . . 10  |-  ( j  =  J  ->  (
( I  =/=  j  ->  ( I X j )  =  .0.  )  <->  ( I  =/=  J  -> 
( I X J )  =  .0.  )
) )
1813, 17rspc2v 3078 . . . . . . . . 9  |-  ( ( I  e.  N  /\  J  e.  N )  ->  ( A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  )  ->  (
I  =/=  J  -> 
( I X J )  =  .0.  )
) )
1918com23 78 . . . . . . . 8  |-  ( ( I  e.  N  /\  J  e.  N )  ->  ( I  =/=  J  ->  ( A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  )  ->  (
I X J )  =  .0.  ) ) )
20193impia 1184 . . . . . . 7  |-  ( ( I  e.  N  /\  J  e.  N  /\  I  =/=  J )  -> 
( A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  )  ->  (
I X J )  =  .0.  ) )
2120com12 31 . . . . . 6  |-  ( A. i  e.  N  A. j  e.  N  (
i  =/=  j  -> 
( i X j )  =  .0.  )  ->  ( ( I  e.  N  /\  J  e.  N  /\  I  =/= 
J )  ->  (
I X J )  =  .0.  ) )
2221a1ii 27 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( X  e.  B  ->  ( A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  )  ->  (
( I  e.  N  /\  J  e.  N  /\  I  =/=  J
)  ->  ( I X J )  =  .0.  ) ) ) )
2322impd 431 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( ( X  e.  B  /\  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i X j )  =  .0.  ) )  -> 
( ( I  e.  N  /\  J  e.  N  /\  I  =/= 
J )  ->  (
I X J )  =  .0.  ) ) )
249, 23syl5bi 217 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( X  e.  D  ->  ( ( I  e.  N  /\  J  e.  N  /\  I  =/= 
J )  ->  (
I X J )  =  .0.  ) ) )
25243impia 1184 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  X  e.  D )  ->  (
( I  e.  N  /\  J  e.  N  /\  I  =/=  J
)  ->  ( I X J )  =  .0.  ) )
2625imp 429 1  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  X  e.  D )  /\  (
I  e.  N  /\  J  e.  N  /\  I  =/=  J ) )  ->  ( I X J )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   A.wral 2714   {crab 2718   ` cfv 5417  (class class class)co 6090   Fincfn 7309   Basecbs 14173   0gc0g 14377   Ringcrg 16644   Mat cmat 18279
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
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-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093
This theorem is referenced by:  dmatmul  30874  dmatsubcl  30875
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