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Theorem maducoeval2 19652
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 2495 . . . . . . . 8  |-  ( m  =  (/)  ->  ( k  e.  m  <->  k  e.  (/) ) )
21ifbid 3931 . . . . . . 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 3928 . . . . . 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 6368 . . . . 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 5882 . . . 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 2436 . . 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 2495 . . . . . . . 8  |-  ( m  =  n  ->  (
k  e.  m  <->  k  e.  n ) )
87ifbid 3931 . . . . . . 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 3928 . . . . . 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 6368 . . . . 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 5882 . . . 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 2436 . . 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 2495 . . . . . . . 8  |-  ( m  =  ( n  u. 
{ r } )  ->  ( k  e.  m  <->  k  e.  ( n  u.  { r } ) ) )
1413ifbid 3931 . . . . . . 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 3928 . . . . . 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 6368 . . . . 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 5882 . . . 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 2436 . . 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 2495 . . . . . . . 8  |-  ( m  =  ( N  \  { H } )  -> 
( k  e.  m  <->  k  e.  ( N  \  { H } ) ) )
2019ifbid 3931 . . . . . . 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 3928 . . . . . 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 6368 . . . . 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 5882 . . . 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 2436 . . 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 19651 . . . . 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 1256 . . . 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 3765 . . . . . . . 8  |-  -.  k  e.  (/)
34 iffalse 3918 . . . . . . . 8  |-  ( -.  k  e.  (/)  ->  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
3533, 34mp1i 13 . . . . . . 7  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
3635ifeq2d 3928 . . . . . 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 6367 . . . . 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 5881 . . . 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 2481 . . 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 2422 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
41 eqid 2422 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
42 eqid 2422 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
43 simpl1l 1056 . . . . . . 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 1030 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  M  e.  B )
4525, 28matrcl 19424 . . . . . . . . . 10  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
4645simpld 460 . . . . . . . . 9  |-  ( M  e.  B  ->  N  e.  Fin )
4744, 46syl 17 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  N  e.  Fin )
4847adantr 466 . . . . . . 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 1029 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  R  e.  CRing
)
5049ad2antrr 730 . . . . . . . . . 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 crngring 17779 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  R  e.  Ring )
5250, 51syl 17 . . . . . . . . 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, 30ring0cl 17790 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
5452, 53syl 17 . . . . . . . 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 1057 . . . . . . . . . . 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 19434 . . . . . . . . . . 11  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
57 elmapi 7498 . . . . . . . . . . 11  |-  ( M  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
5855, 56, 573syl 18 . . . . . . . . . 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 466 . . . . . . . . 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 3587 . . . . . . . . . . . 12  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  r  e.  ( N 
\  { H }
) )
6160ad2antll 733 . . . . . . . . . . 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 3448 . . . . . . . . . 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 466 . . . . . . . . 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 462 . . . . . . . . 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 6452 . . . . . . . 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 3952 . . . . . . 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, 29ringidcl 17789 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
6852, 67syl 17 . . . . . . . 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 3952 . . . . . . 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 1024 . . . . . . . . 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 6451 . . . . . . . . . 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 1201 . . . . . . . . 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 3952 . . . . . . . 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 3952 . . . . . . 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 1009 . . . . . . . 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 6452 . . . . . . 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 1010 . . . . . . 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 4123 . . . . . . . 8  |-  ( r  e.  ( N  \  { H } )  -> 
r  =/=  H )
7961, 78syl 17 . . . . . . 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 19622 . . . . . 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 3954 . . . . . . . . . . . . . . . . 17  |-  if ( -.  l  =  I ,  ( r M l ) ,  .0.  )  =  if (
l  =  I ,  .0.  ,  ( r M l ) )
8281eqcomi 2435 . . . . . . . . . . . . . . . 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 ovif2 6385 . . . . . . . . . . . . . . . 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.  ) )
8576adantr 466 . . . . . . . . . . . . . . . . . . . . 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 ) )
8640, 42, 29ringridm 17793 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .1.  )  =  ( r M I ) )
8752, 85, 86syl2anc 665 . . . . . . . . . . . . . . . . . . . 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 ) )
8887adantr 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 I ) ( .r `  R
)  .1.  )  =  ( r M I ) )
89 oveq2 6310 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  I  ->  (
r M l )  =  ( r M I ) )
9089adantl 467 . . . . . . . . . . . . . . . . . . 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 ) )
9188, 90eqtr4d 2466 . . . . . . . . . . . . . . . . . 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 ) )
9291ifeq1da 3939 . . . . . . . . . . . . . . . . 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.  )
) )
9340, 42, 30ringrz 17806 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .0.  )  =  .0.  )
9452, 85, 93syl2anc 665 . . . . . . . . . . . . . . . . . 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.  )
9594ifeq2d 3928 . . . . . . . . . . . . . . . . 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.  ) )
9692, 95eqtrd 2463 . . . . . . . . . . . . . . . 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.  ) )
9784, 96syl5eq 2475 . . . . . . . . . . . . . . 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.  ) )
9883, 97oveq12d 6320 . . . . . . . . . . . . . 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.  ) ) )
99 ringmnd 17777 . . . . . . . . . . . . . . . 16  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
10052, 99syl 17 . . . . . . . . . . . . . . 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 )
101 id 23 . . . . . . . . . . . . . . . . 17  |-  ( -.  l  =  I  ->  -.  l  =  I
)
102 imnan 423 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  l  =  I  ->  -.  l  =  I )  <->  -.  ( -.  l  =  I  /\  l  =  I
) )
103101, 102mpbi 211 . . . . . . . . . . . . . . . 16  |-  -.  ( -.  l  =  I  /\  l  =  I
)
104103a1i 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 ) )
10540, 30, 41mndifsplit 19648 . . . . . . . . . . . . . . 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.  ) ) )
106100, 65, 104, 105syl3anc 1264 . . . . . . . . . . . . . 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.  )
) )
107 pm2.1 418 . . . . . . . . . . . . . . 15  |-  ( -.  l  =  I  \/  l  =  I )
108 iftrue 3915 . . . . . . . . . . . . . . 15  |-  ( ( -.  l  =  I  \/  l  =  I )  ->  if (
( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( r M l ) )
109107, 108mp1i 13 . . . . . . . . . . . . . 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 ) )
11098, 106, 1093eqtr2d 2469 . . . . . . . . . . . . 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 ) )
1111103adant2 1024 . . . . . . . . . . . 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 ) )
112 oveq1 6309 . . . . . . . . . . . . 13  |-  ( k  =  r  ->  (
k M l )  =  ( r M l ) )
113112eqeq2d 2436 . . . . . . . . . . . 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 ) ) )
114111, 113syl5ibrcom 225 . . . . . . . . . . 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 ) ) )
115114imp 430 . . . . . . . . . 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 ) )
116 iftrue 3915 . . . . . . . . . . 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.  ) ) ) )
117116adantl 467 . . . . . . . . . 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.  ) ) ) )
11879neneqd 2625 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  -.  r  =  H )
1191183ad2ant1 1026 . . . . . . . . . . . . . 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 )
120 eqeq1 2426 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  =  H  <->  r  =  H ) )
121120notbid 295 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  =  H  <->  -.  r  =  H ) )
122119, 121syl5ibrcom 225 . . . . . . . . . . . . 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 ) )
123122imp 430 . . . . . . . . . . . 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 )
124123iffalsed 3920 . . . . . . . . . . 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 ) ) )
125 eldifn 3588 . . . . . . . . . . . . . . . 16  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  -.  r  e.  n
)
126125ad2antll 733 . . . . . . . . . . . . . . 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 )
1271263ad2ant1 1026 . . . . . . . . . . . . . 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
)
128 eleq1 2494 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  e.  n  <->  r  e.  n ) )
129128notbid 295 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  e.  n  <->  -.  r  e.  n ) )
130127, 129syl5ibrcom 225 . . . . . . . . . . . . 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 ) )
131130imp 430 . . . . . . . . . . . 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 )
132131iffalsed 3920 . . . . . . . . . . 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 ) )
133124, 132eqtrd 2463 . . . . . . . . . 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 ) )
134115, 117, 1333eqtr4d 2473 . . . . . . . . 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 ) ) ) )
135 iffalse 3918 . . . . . . . . . 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 ) ) ) )
136135adantl 467 . . . . . . . . 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 ) ) ) )
137134, 136pm2.61dan 798 . . . . . . . 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 ) ) ) )
138137mpt2eq3dva 6366 . . . . . . 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 ) ) ) ) )
139138fveq2d 5882 . . . . . 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 ) ) ) ) ) )
140 neeq2 2707 . . . . . . . . . . . . . . 15  |-  ( k  =  H  ->  (
r  =/=  k  <->  r  =/=  H ) )
141140biimparc 489 . . . . . . . . . . . . . 14  |-  ( ( r  =/=  H  /\  k  =  H )  ->  r  =/=  k )
142141necomd 2695 . . . . . . . . . . . . 13  |-  ( ( r  =/=  H  /\  k  =  H )  ->  k  =/=  r )
143142neneqd 2625 . . . . . . . . . . . 12  |-  ( ( r  =/=  H  /\  k  =  H )  ->  -.  k  =  r )
144143iffalsed 3920 . . . . . . . . . . 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.  ) )
145 iftrue 3915 . . . . . . . . . . . . 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.  ) )
146145adantl 467 . . . . . . . . . . . 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.  ) )
147146ifeq2d 3928 . . . . . . . . . . 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.  ) ) )
148 iftrue 3915 . . . . . . . . . . . 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.  )
)
149148adantl 467 . . . . . . . . . . 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.  ) )
150144, 147, 1493eqtr4d 2473 . . . . . . . . . 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 ) ) ) )
151112ifeq2d 3928 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( r M l ) ) )
152 ssnid 4025 . . . . . . . . . . . . . . . . 17  |-  r  e. 
{ r }
153 elun2 3634 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  { r }  ->  r  e.  ( n  u.  { r } ) )
154152, 153ax-mp 5 . . . . . . . . . . . . . . . 16  |-  r  e.  ( n  u.  {
r } )
155 eleq1 2494 . . . . . . . . . . . . . . . 16  |-  ( k  =  r  ->  (
k  e.  ( n  u.  { r } )  <->  r  e.  ( n  u.  { r } ) ) )
156154, 155mpbiri 236 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  k  e.  ( n  u.  {
r } ) )
157156iftrued 3917 . . . . . . . . . . . . . 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 ) ) )
158 iftrue 3915 . . . . . . . . . . . . . 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 ) ) )
159151, 157, 1583eqtr4rd 2474 . . . . . . . . . . . . 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 ) ) )
160159adantl 467 . . . . . . . . . . . 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 ) ) )
161 iffalse 3918 . . . . . . . . . . . . . 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 ) ) )
162 orc 386 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  n  ->  (
k  e.  n  \/  k  =  r ) )
163 orel2 384 . . . . . . . . . . . . . . . . 17  |-  ( -.  k  =  r  -> 
( ( k  e.  n  \/  k  =  r )  ->  k  e.  n ) )
164162, 163impbid2 207 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  ( k  e.  n  \/  k  =  r ) ) )
165 elun 3606 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( n  u. 
{ r } )  <-> 
( k  e.  n  \/  k  e.  { r } ) )
166 elsn 4010 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  { r }  <-> 
k  =  r )
167166orbi2i 521 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  n  \/  k  e.  { r } )  <->  ( k  e.  n  \/  k  =  r ) )
168165, 167bitr2i 253 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  n  \/  k  =  r )  <-> 
k  e.  ( n  u.  { r } ) )
169164, 168syl6bb 264 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  k  e.  ( n  u. 
{ r } ) ) )
170169ifbid 3931 . . . . . . . . . . . . . 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 ) ) )
171161, 170eqtrd 2463 . . . . . . . . . . . . 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 ) ) )
172171adantl 467 . . . . . . . . . . . 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 ) ) )
173160, 172pm2.61dan 798 . . . . . . . . . . 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 ) ) )
174 iffalse 3918 . . . . . . . . . . . . 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 ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )
175174ifeq2d 3928 . . . . . . . . . . . 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 ) ) ) )
176175adantl 467 . . . . . . . . . . 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 ) ) ) )
177 iffalse 3918 . . . . . . . . . . . 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 ) ) )
178177adantl 467 . . . . . . . . . . 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 ) ) )
179173, 176, 1783eqtr4d 2473 . . . . . . . . . 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 ) ) ) )
180150, 179pm2.61dan 798 . . . . . . . . 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 ) ) ) )
181180mpt2eq3dv 6368 . . . . . . . 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 ) ) ) ) )
182181fveq2d 5882 . . . . . . 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 ) ) ) ) ) )
18379, 182syl 17 . . . . . 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 ) ) ) ) ) )
18480, 139, 1833eqtr3d 2471 . . . . 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 ) ) ) ) ) )
185184eqeq2d 2436 . . . 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 ) ) ) ) ) ) )
186185biimpd 210 . . 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 ) ) ) ) ) ) )
187 difss 3592 . . . 4  |-  ( N 
\  { H }
)  C_  N
188 ssfi 7795 . . . 4  |-  ( ( N  e.  Fin  /\  ( N  \  { H } )  C_  N
)  ->  ( N  \  { H } )  e.  Fin )
18947, 187, 188sylancl 666 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( N  \  { H } )  e.  Fin )
1906, 12, 18, 24, 39, 186, 189findcard2d 7816 . 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 ) ) ) ) ) )
191 iba 505 . . . . . . . 8  |-  ( k  =  H  ->  (
l  =  I  <->  ( l  =  I  /\  k  =  H ) ) )
192191ifbid 3931 . . . . . . 7  |-  ( k  =  H  ->  if ( l  =  I ,  .1.  ,  .0.  )  =  if (
( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
193 iftrue 3915 . . . . . . 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.  )
)
194 iftrue 3915 . . . . . . . 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.  ) )
195194orcs 395 . . . . . . 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.  ) )
196192, 193, 1953eqtr4d 2473 . . . . . 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 ) ) )
197196adantl 467 . . . . 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 ) ) )
198 iffalse 3918 . . . . . . 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 ) ) )
199198adantl 467 . . . . . 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 ) ) )
200 id 23 . . . . . . . . . . 11  |-  ( -.  k  =  H  ->  -.  k  =  H
)
201200neqned 2627 . . . . . . . . . 10  |-  ( -.  k  =  H  -> 
k  =/=  H )
202201anim2i 571 . . . . . . . . 9  |-  ( ( k  e.  N  /\  -.  k  =  H
)  ->  ( k  e.  N  /\  k  =/=  H ) )
203202adantlr 719 . . . . . . . 8  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  (
k  e.  N  /\  k  =/=  H ) )
204 eldifsn 4122 . . . . . . . 8  |-  ( k  e.  ( N  \  { H } )  <->  ( k  e.  N  /\  k  =/=  H ) )
205203, 204sylibr 215 . . . . . . 7  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  k  e.  ( N  \  { H } ) )
206205iftrued 3917 . . . . . 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 ) ) )
207 biorf 406 . . . . . . . 8  |-  ( -.  k  =  H  -> 
( l  =  I  <-> 
( k  =  H  \/  l  =  I ) ) )
208200intnand 924 . . . . . . . . . 10  |-  ( -.  k  =  H  ->  -.  ( l  =  I  /\  k  =  H ) )
209208iffalsed 3920 . . . . . . . . 9  |-  ( -.  k  =  H  ->  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )  =  .0.  )
210209eqcomd 2430 . . . . . . . 8  |-  ( -.  k  =  H  ->  .0.  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )
)
211207, 210ifbieq1d 3932 . . . . . . 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 ) ) )
212211adantl 467 . . . . . 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 ) ) )
213199, 206, 2123eqtrd 2467 . . . . 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 ) ) )
214197, 213pm2.61dan 798 . . . 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 ) ) )
215214mpt2eq3ia 6367 . . 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 ) ) )
216215fveq2i 5881 . 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 ) ) ) )
217190, 216syl6eq 2479 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 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   _Vcvv 3081    \ cdif 3433    u. cun 3434    C_ wss 3436   (/)c0 3761   ifcif 3909   {csn 3996    X. cxp 4848   -->wf 5594   ` cfv 5598  (class class class)co 6302    |-> cmpt2 6304    ^m cmap 7477   Fincfn 7574   Basecbs 15109   +g cplusg 15178   .rcmulr 15179   0gc0g 15326   Mndcmnd 16523   1rcur 17723   Ringcrg 17768   CRingccrg 17769   Mat cmat 19419   maDet cmdat 19596   maAdju cmadu 19644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-ot 4005  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-sup 7959  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-rp 11304  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12657  df-lsw 12658  df-concat 12659  df-s1 12660  df-substr 12661  df-splice 12662  df-reverse 12663  df-s2 12935  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-0g 15328  df-gsum 15329  df-prds 15334  df-pws 15336  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-mhm 16570  df-submnd 16571  df-grp 16661  df-minusg 16662  df-mulg 16664  df-subg 16802  df-ghm 16869  df-gim 16911  df-cntz 16959  df-oppg 16985  df-symg 17007  df-pmtr 17071  df-psgn 17120  df-evpm 17121  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-cring 17771  df-oppr 17839  df-dvdsr 17857  df-unit 17858  df-invr 17888  df-dvr 17899  df-rnghom 17931  df-drng 17965  df-subrg 17994  df-sra 18383  df-rgmod 18384  df-cnfld 18959  df-zring 19027  df-zrh 19062  df-dsmm 19282  df-frlm 19297  df-mat 19420  df-mdet 19597  df-madu 19646
This theorem is referenced by:  madutpos  19654
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