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Theorem dmatelnd 18867
Description: An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
dmatid.a  |-  A  =  ( N Mat  R )
dmatid.b  |-  B  =  ( Base `  A
)
dmatid.0  |-  .0.  =  ( 0g `  R )
dmatid.d  |-  D  =  ( N DMat  R )
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.  )

Proof of Theorem dmatelnd
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmatid.a . . . . 5  |-  A  =  ( N Mat  R )
2 dmatid.b . . . . 5  |-  B  =  ( Base `  A
)
3 dmatid.0 . . . . 5  |-  .0.  =  ( 0g `  R )
4 dmatid.d . . . . 5  |-  D  =  ( N DMat  R )
51, 2, 3, 4dmatel 18864 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( X  e.  D  <->  ( X  e.  B  /\  A. i  e.  N  A. j  e.  N  (
i  =/=  j  -> 
( i X j )  =  .0.  )
) ) )
6 neeq1 2748 . . . . . . . . . . 11  |-  ( i  =  I  ->  (
i  =/=  j  <->  I  =/=  j ) )
7 oveq1 6302 . . . . . . . . . . . 12  |-  ( i  =  I  ->  (
i X j )  =  ( I X j ) )
87eqeq1d 2469 . . . . . . . . . . 11  |-  ( i  =  I  ->  (
( i X j )  =  .0.  <->  ( I X j )  =  .0.  ) )
96, 8imbi12d 320 . . . . . . . . . 10  |-  ( i  =  I  ->  (
( i  =/=  j  ->  ( i X j )  =  .0.  )  <->  ( I  =/=  j  -> 
( I X j )  =  .0.  )
) )
10 neeq2 2750 . . . . . . . . . . 11  |-  ( j  =  J  ->  (
I  =/=  j  <->  I  =/=  J ) )
11 oveq2 6303 . . . . . . . . . . . 12  |-  ( j  =  J  ->  (
I X j )  =  ( I X J ) )
1211eqeq1d 2469 . . . . . . . . . . 11  |-  ( j  =  J  ->  (
( I X j )  =  .0.  <->  ( I X J )  =  .0.  ) )
1310, 12imbi12d 320 . . . . . . . . . 10  |-  ( j  =  J  ->  (
( I  =/=  j  ->  ( I X j )  =  .0.  )  <->  ( I  =/=  J  -> 
( I X J )  =  .0.  )
) )
149, 13rspc2v 3228 . . . . . . . . 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.  )
) )
1514com23 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.  ) ) )
16153impia 1193 . . . . . . 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.  ) )
1716com12 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.  ) )
1817a1ii 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.  ) ) ) )
1918impd 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.  ) ) )
205, 19sylbid 215 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( X  e.  D  ->  ( ( I  e.  N  /\  J  e.  N  /\  I  =/= 
J )  ->  (
I X J )  =  .0.  ) ) )
21203impia 1193 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  X  e.  D )  ->  (
( I  e.  N  /\  J  e.  N  /\  I  =/=  J
)  ->  ( I X J )  =  .0.  ) )
2221imp 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   ` cfv 5594  (class class class)co 6295   Fincfn 7528   Basecbs 14507   0gc0g 14712   Ringcrg 17070   Mat cmat 18778   DMat cdmat 18859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-dmat 18861
This theorem is referenced by:  dmatmul  18868  dmatsubcl  18869
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