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Theorem maducoeval2 18902
Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 17-Jul-2018.)
Hypotheses
Ref Expression
madufval.a  |-  A  =  ( N Mat  R )
madufval.d  |-  D  =  ( N maDet  R )
madufval.j  |-  J  =  ( N maAdju  R )
madufval.b  |-  B  =  ( Base `  A
)
madufval.o  |-  .1.  =  ( 1r `  R )
madufval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
maducoeval2  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
Distinct variable groups:    k, N, l    R, k, l    k, M, l    k, I, l   
k, H, l    B, k, l    .0. , k    .1. , k
Allowed substitution hints:    A( k, l)    D( k, l)    .1. ( l)    J( k, l)    .0. ( l)

Proof of Theorem maducoeval2
Dummy variables  n  r  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2533 . . . . . . . 8  |-  ( m  =  (/)  ->  ( k  e.  m  <->  k  e.  (/) ) )
21ifbid 3954 . . . . . . 7  |-  ( m  =  (/)  ->  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
32ifeq2d 3951 . . . . . 6  |-  ( m  =  (/)  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
43mpt2eq3dv 6338 . . . . 5  |-  ( m  =  (/)  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )
54fveq2d 5861 . . . 4  |-  ( m  =  (/)  ->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
65eqeq2d 2474 . . 3  |-  ( m  =  (/)  ->  ( ( I ( J `  M ) H )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )  <->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
7 eleq2 2533 . . . . . . . 8  |-  ( m  =  n  ->  (
k  e.  m  <->  k  e.  n ) )
87ifbid 3954 . . . . . . 7  |-  ( m  =  n  ->  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )
98ifeq2d 3951 . . . . . 6  |-  ( m  =  n  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
109mpt2eq3dv 6338 . . . . 5  |-  ( m  =  n  ->  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )
1110fveq2d 5861 . . . 4  |-  ( m  =  n  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
1211eqeq2d 2474 . . 3  |-  ( m  =  n  ->  (
( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  <-> 
( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
13 eleq2 2533 . . . . . . . 8  |-  ( m  =  ( n  u. 
{ r } )  ->  ( k  e.  m  <->  k  e.  ( n  u.  { r } ) ) )
1413ifbid 3954 . . . . . . 7  |-  ( m  =  ( n  u. 
{ r } )  ->  if ( k  e.  m ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  if ( k  e.  ( n  u. 
{ r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
1514ifeq2d 3951 . . . . . 6  |-  ( m  =  ( n  u. 
{ r } )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
1615mpt2eq3dv 6338 . . . . 5  |-  ( m  =  ( n  u. 
{ r } )  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  {
r } ) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )
1716fveq2d 5861 . . . 4  |-  ( m  =  ( n  u. 
{ r } )  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
1817eqeq2d 2474 . . 3  |-  ( m  =  ( n  u. 
{ r } )  ->  ( ( I ( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  <-> 
( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
19 eleq2 2533 . . . . . . . 8  |-  ( m  =  ( N  \  { H } )  -> 
( k  e.  m  <->  k  e.  ( N  \  { H } ) ) )
2019ifbid 3954 . . . . . . 7  |-  ( m  =  ( N  \  { H } )  ->  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  if ( k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
2120ifeq2d 3951 . . . . . 6  |-  ( m  =  ( N  \  { H } )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
2221mpt2eq3dv 6338 . . . . 5  |-  ( m  =  ( N  \  { H } )  -> 
( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )
2322fveq2d 5861 . . . 4  |-  ( m  =  ( N  \  { H } )  -> 
( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
2423eqeq2d 2474 . . 3  |-  ( m  =  ( N  \  { H } )  -> 
( ( I ( J `  M ) H )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  m ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  <-> 
( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
25 madufval.a . . . . . 6  |-  A  =  ( N Mat  R )
26 madufval.d . . . . . 6  |-  D  =  ( N maDet  R )
27 madufval.j . . . . . 6  |-  J  =  ( N maAdju  R )
28 madufval.b . . . . . 6  |-  B  =  ( Base `  A
)
29 madufval.o . . . . . 6  |-  .1.  =  ( 1r `  R )
30 madufval.z . . . . . 6  |-  .0.  =  ( 0g `  R )
3125, 26, 27, 28, 29, 30maducoeval 18901 . . . . 5  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
32313adant1l 1215 . . . 4  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
33 noel 3782 . . . . . . . 8  |-  -.  k  e.  (/)
34 iffalse 3941 . . . . . . . 8  |-  ( -.  k  e.  (/)  ->  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
3533, 34mp1i 12 . . . . . . 7  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
3635ifeq2d 3951 . . . . . 6  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
3736mpt2eq3ia 6337 . . . . 5  |-  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )  =  ( k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
3837fveq2i 5860 . . . 4  |-  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
3932, 38syl6eqr 2519 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) )
40 eqid 2460 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
41 eqid 2460 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
42 eqid 2460 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
43 simpl1l 1042 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  R  e.  CRing )
44 simp1r 1016 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  M  e.  B )
4525, 28matrcl 18674 . . . . . . . . . 10  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
4645simpld 459 . . . . . . . . 9  |-  ( M  e.  B  ->  N  e.  Fin )
4744, 46syl 16 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  N  e.  Fin )
4847adantr 465 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  N  e.  Fin )
49 simp1l 1015 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  R  e.  CRing
)
5049ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  R  e.  CRing )
51 crngrng 16989 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  R  e.  Ring )
5250, 51syl 16 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  R  e.  Ring )
5340, 30rng0cl 17000 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
5452, 53syl 16 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  .0.  e.  ( Base `  R ) )
55 simpl1r 1043 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  M  e.  B
)
5625, 40, 28matbas2i 18684 . . . . . . . . . . 11  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
57 elmapi 7430 . . . . . . . . . . 11  |-  ( M  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
5855, 56, 573syl 20 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
5958adantr 465 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  M : ( N  X.  N ) --> (
Base `  R )
)
60 eldifi 3619 . . . . . . . . . . . 12  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  r  e.  ( N 
\  { H }
) )
6160ad2antll 728 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  r  e.  ( N  \  { H } ) )
6261eldifad 3481 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  r  e.  N
)
6362adantr 465 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  r  e.  N )
64 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  l  e.  N )
6559, 63, 64fovrnd 6422 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( r M l )  e.  ( Base `  R ) )
6654, 65ifcld 3975 . . . . . . 7  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( l  =  I ,  .0.  , 
( r M l ) )  e.  (
Base `  R )
)
6740, 29rngidcl 16999 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
6852, 67syl 16 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  .1.  e.  ( Base `  R ) )
6968, 54ifcld 3975 . . . . . . 7  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( l  =  I ,  .1.  ,  .0.  )  e.  ( Base `  R ) )
70543adant2 1010 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  .0.  e.  ( Base `  R ) )
7158fovrnda 6421 . . . . . . . . . 10  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  ( k  e.  N  /\  l  e.  N
) )  ->  (
k M l )  e.  ( Base `  R
) )
72713impb 1187 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k M l )  e.  ( Base `  R ) )
7370, 72ifcld 3975 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  if ( l  =  I ,  .0.  , 
( k M l ) )  e.  (
Base `  R )
)
7473, 72ifcld 3975 . . . . . . 7  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  e.  ( Base `  R
) )
75 simpl2 995 . . . . . . . 8  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  I  e.  N
)
7658, 62, 75fovrnd 6422 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( r M I )  e.  (
Base `  R )
)
77 simpl3 996 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  H  e.  N
)
78 eldifsni 4146 . . . . . . . 8  |-  ( r  e.  ( N  \  { H } )  -> 
r  =/=  H )
7961, 78syl 16 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  r  =/=  H
)
8026, 40, 41, 42, 43, 48, 66, 69, 74, 76, 62, 77, 79mdetero 18872 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
81 ifnot 3977 . . . . . . . . . . . . . . . . 17  |-  if ( -.  l  =  I ,  ( r M l ) ,  .0.  )  =  if (
l  =  I ,  .0.  ,  ( r M l ) )
8281eqcomi 2473 . . . . . . . . . . . . . . . 16  |-  if ( l  =  I ,  .0.  ,  ( r M l ) )  =  if ( -.  l  =  I ,  ( r M l ) ,  .0.  )
8382a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( l  =  I ,  .0.  , 
( r M l ) )  =  if ( -.  l  =  I ,  ( r M l ) ,  .0.  ) )
84 oveq2 6283 . . . . . . . . . . . . . . . . 17  |-  ( if ( l  =  I ,  .1.  ,  .0.  )  =  .1.  ->  ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  )
)  =  ( ( r M I ) ( .r `  R
)  .1.  ) )
85 oveq2 6283 . . . . . . . . . . . . . . . . 17  |-  ( if ( l  =  I ,  .1.  ,  .0.  )  =  .0.  ->  ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  )
)  =  ( ( r M I ) ( .r `  R
)  .0.  ) )
8684, 85ifsb 3945 . . . . . . . . . . . . . . . 16  |-  ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) )  =  if ( l  =  I ,  ( ( r M I ) ( .r `  R
)  .1.  ) ,  ( ( r M I ) ( .r
`  R )  .0.  ) )
8776adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( r M I )  e.  ( Base `  R ) )
8840, 42, 29rngridm 17003 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .1.  )  =  ( r M I ) )
8952, 87, 88syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( ( r M I ) ( .r
`  R )  .1.  )  =  ( r M I ) )
9089adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  /\  l  =  I
)  ->  ( (
r M I ) ( .r `  R
)  .1.  )  =  ( r M I ) )
91 oveq2 6283 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  I  ->  (
r M l )  =  ( r M I ) )
9291adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  /\  l  =  I
)  ->  ( r M l )  =  ( r M I ) )
9390, 92eqtr4d 2504 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  /\  l  =  I
)  ->  ( (
r M I ) ( .r `  R
)  .1.  )  =  ( r M l ) )
9493ifeq1da 3962 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( l  =  I ,  ( ( r M I ) ( .r `  R
)  .1.  ) ,  ( ( r M I ) ( .r
`  R )  .0.  ) )  =  if ( l  =  I ,  ( r M l ) ,  ( ( r M I ) ( .r `  R )  .0.  )
) )
9540, 42, 30rngrz 17016 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .0.  )  =  .0.  )
9652, 87, 95syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( ( r M I ) ( .r
`  R )  .0.  )  =  .0.  )
9796ifeq2d 3951 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( l  =  I ,  ( r M l ) ,  ( ( r M I ) ( .r
`  R )  .0.  ) )  =  if ( l  =  I ,  ( r M l ) ,  .0.  ) )
9894, 97eqtrd 2501 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( l  =  I ,  ( ( r M I ) ( .r `  R
)  .1.  ) ,  ( ( r M I ) ( .r
`  R )  .0.  ) )  =  if ( l  =  I ,  ( r M l ) ,  .0.  ) )
9986, 98syl5eq 2513 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( ( r M I ) ( .r
`  R ) if ( l  =  I ,  .1.  ,  .0.  ) )  =  if ( l  =  I ,  ( r M l ) ,  .0.  ) )
10083, 99oveq12d 6293 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( if ( -.  l  =  I ,  ( r M l ) ,  .0.  ) ( +g  `  R
) if ( l  =  I ,  ( r M l ) ,  .0.  ) ) )
101 rngmnd 16988 . . . . . . . . . . . . . . . 16  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
10252, 101syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  R  e.  Mnd )
103 id 22 . . . . . . . . . . . . . . . . 17  |-  ( -.  l  =  I  ->  -.  l  =  I
)
104 imnan 422 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  l  =  I  ->  -.  l  =  I )  <->  -.  ( -.  l  =  I  /\  l  =  I
) )
105103, 104mpbi 208 . . . . . . . . . . . . . . . 16  |-  -.  ( -.  l  =  I  /\  l  =  I
)
106105a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  -.  ( -.  l  =  I  /\  l  =  I ) )
10740, 30, 41mndifsplit 18898 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Mnd  /\  ( r M l )  e.  ( Base `  R )  /\  -.  ( -.  l  =  I  /\  l  =  I ) )  ->  if ( ( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( if ( -.  l  =  I ,  ( r M l ) ,  .0.  )
( +g  `  R ) if ( l  =  I ,  ( r M l ) ,  .0.  ) ) )
108102, 65, 106, 107syl3anc 1223 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( ( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( if ( -.  l  =  I ,  ( r M l ) ,  .0.  ) ( +g  `  R ) if ( l  =  I ,  ( r M l ) ,  .0.  )
) )
109 pm2.1 417 . . . . . . . . . . . . . . 15  |-  ( -.  l  =  I  \/  l  =  I )
110 iftrue 3938 . . . . . . . . . . . . . . 15  |-  ( ( -.  l  =  I  \/  l  =  I )  ->  if (
( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( r M l ) )
111109, 110mp1i 12 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( ( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( r M l ) )
112100, 108, 1113eqtr2d 2507 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( r M l ) )
1131123adant2 1010 . . . . . . . . . . . 12  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( r M l ) )
114 oveq1 6282 . . . . . . . . . . . . 13  |-  ( k  =  r  ->  (
k M l )  =  ( r M l ) )
115114eqeq2d 2474 . . . . . . . . . . . 12  |-  ( k  =  r  ->  (
( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( k M l )  <->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( r M l ) ) )
116113, 115syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k  =  r  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( k M l ) ) )
117116imp 429 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( k M l ) )
118 iftrue 3938 . . . . . . . . . . 11  |-  ( k  =  r  ->  if ( k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )  =  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) ) )
119118adantl 466 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  if (
k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) )
12079neneqd 2662 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  -.  r  =  H )
1211203ad2ant1 1012 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  -.  r  =  H )
122 eqeq1 2464 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  =  H  <->  r  =  H ) )
123122notbid 294 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  =  H  <->  -.  r  =  H ) )
124121, 123syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k  =  r  ->  -.  k  =  H ) )
125124imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  -.  k  =  H )
126 iffalse 3941 . . . . . . . . . . . 12  |-  ( -.  k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )
127125, 126syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  if (
k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )
128 eldifn 3620 . . . . . . . . . . . . . . . 16  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  -.  r  e.  n
)
129128ad2antll 728 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  -.  r  e.  n )
1301293ad2ant1 1012 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  -.  r  e.  n
)
131 eleq1 2532 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  e.  n  <->  r  e.  n ) )
132131notbid 294 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  e.  n  <->  -.  r  e.  n ) )
133130, 132syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k  =  r  ->  -.  k  e.  n ) )
134133imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  -.  k  e.  n )
135 iffalse 3941 . . . . . . . . . . . 12  |-  ( -.  k  e.  n  ->  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
136134, 135syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
137127, 136eqtrd 2501 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  if (
k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  ( k M l ) )
138117, 119, 1373eqtr4d 2511 . . . . . . . . 9  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  if (
k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
139 iffalse 3941 . . . . . . . . . 10  |-  ( -.  k  =  r  ->  if ( k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
140139adantl 466 . . . . . . . . 9  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  -.  k  =  r )  ->  if (
k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
141138, 140pm2.61dan 789 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  if ( k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
142141mpt2eq3dva 6336 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  r ,  ( if ( l  =  I ,  .0.  , 
( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )
143142fveq2d 5861 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) )
144 neeq2 2743 . . . . . . . . . . . . . . 15  |-  ( k  =  H  ->  (
r  =/=  k  <->  r  =/=  H ) )
145144biimparc 487 . . . . . . . . . . . . . 14  |-  ( ( r  =/=  H  /\  k  =  H )  ->  r  =/=  k )
146145necomd 2731 . . . . . . . . . . . . 13  |-  ( ( r  =/=  H  /\  k  =  H )  ->  k  =/=  r )
147146neneqd 2662 . . . . . . . . . . . 12  |-  ( ( r  =/=  H  /\  k  =  H )  ->  -.  k  =  r )
148 iffalse 3941 . . . . . . . . . . . 12  |-  ( -.  k  =  r  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( l  =  I ,  .1.  ,  .0.  ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
149147, 148syl 16 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( l  =  I ,  .1.  ,  .0.  ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
150 iftrue 3938 . . . . . . . . . . . . 13  |-  ( k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
151150adantl 466 . . . . . . . . . . . 12  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
152151ifeq2d 3951 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( l  =  I ,  .1.  ,  .0.  ) ) )
153 iftrue 3938 . . . . . . . . . . . 12  |-  ( k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  )
)
154153adantl 466 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  {
r } ) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
155149, 152, 1543eqtr4d 2511 . . . . . . . . . 10  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  {
r } ) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
156114ifeq2d 3951 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( r M l ) ) )
157 vex 3109 . . . . . . . . . . . . . . . . . 18  |-  r  e. 
_V
158157snid 4048 . . . . . . . . . . . . . . . . 17  |-  r  e. 
{ r }
159 elun2 3665 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  { r }  ->  r  e.  ( n  u.  { r } ) )
160158, 159ax-mp 5 . . . . . . . . . . . . . . . 16  |-  r  e.  ( n  u.  {
r } )
161 eleq1 2532 . . . . . . . . . . . . . . . 16  |-  ( k  =  r  ->  (
k  e.  ( n  u.  { r } )  <->  r  e.  ( n  u.  { r } ) ) )
162160, 161mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  k  e.  ( n  u.  {
r } ) )
163 iftrue 3938 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( n  u. 
{ r } )  ->  if ( k  e.  ( n  u. 
{ r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( k M l ) ) )
164162, 163syl 16 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( k M l ) ) )
165 iftrue 3938 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .0.  ,  ( r M l ) ) )
166156, 164, 1653eqtr4rd 2512 . . . . . . . . . . . . 13  |-  ( k  =  r  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
167166adantl 466 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  H  /\  -.  k  =  H )  /\  k  =  r )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
168 iffalse 3941 . . . . . . . . . . . . . 14  |-  ( -.  k  =  r  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )
169 orc 385 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  n  ->  (
k  e.  n  \/  k  =  r ) )
170 orel2 383 . . . . . . . . . . . . . . . . 17  |-  ( -.  k  =  r  -> 
( ( k  e.  n  \/  k  =  r )  ->  k  e.  n ) )
171169, 170impbid2 204 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  ( k  e.  n  \/  k  =  r ) ) )
172 elun 3638 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( n  u. 
{ r } )  <-> 
( k  e.  n  \/  k  e.  { r } ) )
173 elsn 4034 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  { r }  <-> 
k  =  r )
174173orbi2i 519 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  n  \/  k  e.  { r } )  <->  ( k  e.  n  \/  k  =  r ) )
175172, 174bitr2i 250 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  n  \/  k  =  r )  <-> 
k  e.  ( n  u.  { r } ) )
176171, 175syl6bb 261 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  k  e.  ( n  u. 
{ r } ) ) )
177176ifbid 3954 . . . . . . . . . . . . . 14  |-  ( -.  k  =  r  ->  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
178168, 177eqtrd 2501 . . . . . . . . . . . . 13  |-  ( -.  k  =  r  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
179178adantl 466 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  H  /\  -.  k  =  H )  /\  -.  k  =  r )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
180167, 179pm2.61dan 789 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  -.  k  =  H
)  ->  if (
k  =  r ,  if ( l  =  I ,  .0.  , 
( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
181126ifeq2d 3951 . . . . . . . . . . . 12  |-  ( -.  k  =  H  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
182181adantl 466 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  -.  k  =  H
)  ->  if (
k  =  r ,  if ( l  =  I ,  .0.  , 
( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
183 iffalse 3941 . . . . . . . . . . . 12  |-  ( -.  k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
184183adantl 466 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  -.  k  =  H
)  ->  if (
k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u. 
{ r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
185180, 182, 1843eqtr4d 2511 . . . . . . . . . 10  |-  ( ( r  =/=  H  /\  -.  k  =  H
)  ->  if (
k  =  r ,  if ( l  =  I ,  .0.  , 
( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  {
r } ) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
186155, 185pm2.61dan 789 . . . . . . . . 9  |-  ( r  =/=  H  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  {
r } ) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )
187186mpt2eq3dv 6338 . . . . . . . 8  |-  ( r  =/=  H  ->  (
k  e.  N , 
l  e.  N  |->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )
188187fveq2d 5861 . . . . . . 7  |-  ( r  =/=  H  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
18979, 188syl 16 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
19080, 143, 1893eqtr3d 2509 . . . . 5  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
191190eqeq2d 2474 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( ( I ( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  <-> 
( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
192191biimpd 207 . . 3  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( ( I ( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  ->  ( I ( J `  M ) H )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
193 difss 3624 . . . 4  |-  ( N 
\  { H }
)  C_  N
194 ssfi 7730 . . . 4  |-  ( ( N  e.  Fin  /\  ( N  \  { H } )  C_  N
)  ->  ( N  \  { H } )  e.  Fin )
19547, 193, 194sylancl 662 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( N  \  { H } )  e.  Fin )
1966, 12, 18, 24, 39, 192, 195findcard2d 7751 . 2  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
197 iba 503 . . . . . . . 8  |-  ( k  =  H  ->  (
l  =  I  <->  ( l  =  I  /\  k  =  H ) ) )
198197ifbid 3954 . . . . . . 7  |-  ( k  =  H  ->  if ( l  =  I ,  .1.  ,  .0.  )  =  if (
( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
199 iftrue 3938 . . . . . . 7  |-  ( k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  )
)
200 iftrue 3938 . . . . . . . 8  |-  ( ( k  =  H  \/  l  =  I )  ->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) )  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
201200orcs 394 . . . . . . 7  |-  ( k  =  H  ->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) )  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
202198, 199, 2013eqtr4d 2511 . . . . . 6  |-  ( k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
203202adantl 466 . . . . 5  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
204 iffalse 3941 . . . . . . 7  |-  ( -.  k  =  H  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
205204adantl 466 . . . . . 6  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
206 id 22 . . . . . . . . . . 11  |-  ( -.  k  =  H  ->  -.  k  =  H
)
207206neqned 2663 . . . . . . . . . 10  |-  ( -.  k  =  H  -> 
k  =/=  H )
208207anim2i 569 . . . . . . . . 9  |-  ( ( k  e.  N  /\  -.  k  =  H
)  ->  ( k  e.  N  /\  k  =/=  H ) )
209208adantlr 714 . . . . . . . 8  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  (
k  e.  N  /\  k  =/=  H ) )
210 eldifsn 4145 . . . . . . . 8  |-  ( k  e.  ( N  \  { H } )  <->  ( k  e.  N  /\  k  =/=  H ) )
211209, 210sylibr 212 . . . . . . 7  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  k  e.  ( N  \  { H } ) )
212 iftrue 3938 . . . . . . 7  |-  ( k  e.  ( N  \  { H } )  ->  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( k M l ) ) )
213211, 212syl 16 . . . . . 6  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( k M l ) ) )
214 biorf 405 . . . . . . . 8  |-  ( -.  k  =  H  -> 
( l  =  I  <-> 
( k  =  H  \/  l  =  I ) ) )
215206intnand 909 . . . . . . . . . 10  |-  ( -.  k  =  H  ->  -.  ( l  =  I  /\  k  =  H ) )
216 iffalse 3941 . . . . . . . . . 10  |-  ( -.  ( l  =  I  /\  k  =  H )  ->  if (
( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )  =  .0.  )
217215, 216syl 16 . . . . . . . . 9  |-  ( -.  k  =  H  ->  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )  =  .0.  )
218217eqcomd 2468 . . . . . . . 8  |-  ( -.  k  =  H  ->  .0.  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )
)
219 eqidd 2461 . . . . . . . 8  |-  ( -.  k  =  H  -> 
( k M l )  =  ( k M l ) )
220214, 218, 219ifbieq12d 3959 . . . . . . 7  |-  ( -.  k  =  H  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
221220adantl 466 . . . . . 6  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
222205, 213, 2213eqtrd 2505 . . . . 5  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
223203, 222pm2.61dan 789 . . . 4  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
224223mpt2eq3ia 6337 . . 3  |-  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
225224fveq2i 5860 . 2  |-  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
226196, 225syl6eq 2517 1  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    \ cdif 3466    u. cun 3467    C_ wss 3469   (/)c0 3778   ifcif 3932   {csn 4020    X. cxp 4990   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    ^m cmap 7410   Fincfn 7506   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   0gc0g 14684   Mndcmnd 15715   1rcur 16936   Ringcrg 16979   CRingccrg 16980   Mat cmat 18669   maDet cmdat 18846   maAdju cmadu 18894
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  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-fal 1380  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-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  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-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  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-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-substr 12499  df-splice 12500  df-reverse 12501  df-s2 12763  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-0g 14686  df-gsum 14687  df-prds 14692  df-pws 14694  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-mulg 15854  df-subg 15986  df-ghm 16053  df-gim 16095  df-cntz 16143  df-oppg 16169  df-symg 16191  df-pmtr 16256  df-psgn 16305  df-evpm 16306  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-rnghom 17141  df-drng 17174  df-subrg 17203  df-sra 17594  df-rgmod 17595  df-cnfld 18185  df-zring 18250  df-zrh 18301  df-dsmm 18523  df-frlm 18538  df-mat 18670  df-mdet 18847  df-madu 18896
This theorem is referenced by:  madutpos  18904
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