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Theorem maducoeval2 19742
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 2538 . . . . . . . 8  |-  ( m  =  (/)  ->  ( k  e.  m  <->  k  e.  (/) ) )
21ifbid 3894 . . . . . . 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 3891 . . . . . 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 6376 . . . . 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 5883 . . . 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 2481 . . 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 2538 . . . . . . . 8  |-  ( m  =  n  ->  (
k  e.  m  <->  k  e.  n ) )
87ifbid 3894 . . . . . . 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 3891 . . . . . 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 6376 . . . . 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 5883 . . . 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 2481 . . 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 2538 . . . . . . . 8  |-  ( m  =  ( n  u. 
{ r } )  ->  ( k  e.  m  <->  k  e.  ( n  u.  { r } ) ) )
1413ifbid 3894 . . . . . . 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 3891 . . . . . 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 6376 . . . . 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 5883 . . . 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 2481 . . 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 2538 . . . . . . . 8  |-  ( m  =  ( N  \  { H } )  -> 
( k  e.  m  <->  k  e.  ( N  \  { H } ) ) )
2019ifbid 3894 . . . . . . 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 3891 . . . . . 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 6376 . . . . 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 5883 . . . 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 2481 . . 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 19741 . . . . 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 1284 . . . 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 3726 . . . . . . . 8  |-  -.  k  e.  (/)
34 iffalse 3881 . . . . . . . 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 3891 . . . . . 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 6375 . . . . 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 5882 . . . 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 2523 . . 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 2471 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
41 eqid 2471 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
42 eqid 2471 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
43 simpl1l 1081 . . . . . . 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 1055 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  M  e.  B )
4525, 28matrcl 19514 . . . . . . . . . 10  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
4645simpld 466 . . . . . . . . 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 472 . . . . . . 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 1054 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  R  e.  CRing
)
5049ad2antrr 740 . . . . . . . . . 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 17869 . . . . . . . . . 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 17880 . . . . . . . . 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 1082 . . . . . . . . . . 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 19524 . . . . . . . . . . 11  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
57 elmapi 7511 . . . . . . . . . . 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 472 . . . . . . . . 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 3544 . . . . . . . . . . . 12  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  r  e.  ( N 
\  { H }
) )
6160ad2antll 743 . . . . . . . . . . 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 3402 . . . . . . . . . 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 472 . . . . . . . . 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 468 . . . . . . . . 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 6460 . . . . . . . 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 3915 . . . . . . 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 17879 . . . . . . . . 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 3915 . . . . . . 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 1049 . . . . . . . . 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 6459 . . . . . . . . . 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 1227 . . . . . . . . 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 3915 . . . . . . . 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 3915 . . . . . . 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 1034 . . . . . . . 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 6460 . . . . . . 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 1035 . . . . . . 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 4089 . . . . . . . 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 19712 . . . . . 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 3917 . . . . . . . . . . . . . . . . 17  |-  if ( -.  l  =  I ,  ( r M l ) ,  .0.  )  =  if (
l  =  I ,  .0.  ,  ( r M l ) )
8281eqcomi 2480 . . . . . . . . . . . . . . . 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 6393 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . 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 17883 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .1.  )  =  ( r M I ) )
8752, 85, 86syl2anc 673 . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  I  ->  (
r M l )  =  ( r M I ) )
9089adantl 473 . . . . . . . . . . . . . . . . . . 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 2508 . . . . . . . . . . . . . . . . . 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 3902 . . . . . . . . . . . . . . . . 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 17896 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .0.  )  =  .0.  )
9452, 85, 93syl2anc 673 . . . . . . . . . . . . . . . . . 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 3891 . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . . 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 6326 . . . . . . . . . . . . . 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 17867 . . . . . . . . . . . . . . . 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 22 . . . . . . . . . . . . . . . . 17  |-  ( -.  l  =  I  ->  -.  l  =  I
)
102 imnan 429 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  l  =  I  ->  -.  l  =  I )  <->  -.  ( -.  l  =  I  /\  l  =  I
) )
103101, 102mpbi 213 . . . . . . . . . . . . . . . 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 19738 . . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . . 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 pm2.1 424 . . . . . . . . . . . . . . 15  |-  ( -.  l  =  I  \/  l  =  I )
108 iftrue 3878 . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . 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 1049 . . . . . . . . . . . 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 6315 . . . . . . . . . . . . 13  |-  ( k  =  r  ->  (
k M l )  =  ( r M l ) )
113112eqeq2d 2481 . . . . . . . . . . . 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 230 . . . . . . . . . . 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 436 . . . . . . . . . 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 3878 . . . . . . . . . . 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 473 . . . . . . . . . 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 2648 . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  =  H  <->  r  =  H ) )
121120notbid 301 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  =  H  <->  -.  r  =  H ) )
122119, 121syl5ibrcom 230 . . . . . . . . . . . . 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 436 . . . . . . . . . . . 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 3883 . . . . . . . . . . 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 3545 . . . . . . . . . . . . . . . 16  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  -.  r  e.  n
)
126125ad2antll 743 . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . 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 2537 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  e.  n  <->  r  e.  n ) )
129128notbid 301 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  e.  n  <->  -.  r  e.  n ) )
130127, 129syl5ibrcom 230 . . . . . . . . . . . . 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 436 . . . . . . . . . . . 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 3883 . . . . . . . . . . 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 2505 . . . . . . . . . 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 2515 . . . . . . . . 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 3881 . . . . . . . . . 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 473 . . . . . . . . 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 808 . . . . . . . 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 6374 . . . . . . 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 5883 . . . . . 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 2706 . . . . . . . . . . . . . . 15  |-  ( k  =  H  ->  (
r  =/=  k  <->  r  =/=  H ) )
141140biimparc 495 . . . . . . . . . . . . . 14  |-  ( ( r  =/=  H  /\  k  =  H )  ->  r  =/=  k )
142141necomd 2698 . . . . . . . . . . . . 13  |-  ( ( r  =/=  H  /\  k  =  H )  ->  k  =/=  r )
143142neneqd 2648 . . . . . . . . . . . 12  |-  ( ( r  =/=  H  /\  k  =  H )  ->  -.  k  =  r )
144143iffalsed 3883 . . . . . . . . . . 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 3878 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 3891 . . . . . . . . . . 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 3878 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 2515 . . . . . . . . . 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 3891 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( r M l ) ) )
152 ssnid 3989 . . . . . . . . . . . . . . . . 17  |-  r  e. 
{ r }
153 elun2 3593 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  { r }  ->  r  e.  ( n  u.  { r } ) )
154152, 153ax-mp 5 . . . . . . . . . . . . . . . 16  |-  r  e.  ( n  u.  {
r } )
155 eleq1 2537 . . . . . . . . . . . . . . . 16  |-  ( k  =  r  ->  (
k  e.  ( n  u.  { r } )  <->  r  e.  ( n  u.  { r } ) ) )
156154, 155mpbiri 241 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  k  e.  ( n  u.  {
r } ) )
157156iftrued 3880 . . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . . 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 2516 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 3881 . . . . . . . . . . . . . 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 392 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  n  ->  (
k  e.  n  \/  k  =  r ) )
163 orel2 390 . . . . . . . . . . . . . . . . 17  |-  ( -.  k  =  r  -> 
( ( k  e.  n  \/  k  =  r )  ->  k  e.  n ) )
164162, 163impbid2 209 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  ( k  e.  n  \/  k  =  r ) ) )
165 elun 3565 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( n  u. 
{ r } )  <-> 
( k  e.  n  \/  k  e.  { r } ) )
166 elsn 3973 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  { r }  <-> 
k  =  r )
167166orbi2i 528 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  n  \/  k  e.  { r } )  <->  ( k  e.  n  \/  k  =  r ) )
168165, 167bitr2i 258 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  n  \/  k  =  r )  <-> 
k  e.  ( n  u.  { r } ) )
169164, 168syl6bb 269 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  k  e.  ( n  u. 
{ r } ) ) )
170169ifbid 3894 . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 808 . . . . . . . . . . 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 3881 . . . . . . . . . . . . 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 3891 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 3881 . . . . . . . . . . . 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 473 . . . . . . . . . . 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 2515 . . . . . . . . . 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 808 . . . . . . . . 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 6376 . . . . . . . 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 5883 . . . . . . 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 2513 . . . . 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 2481 . . . 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 212 . . 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 3549 . . . 4  |-  ( N 
\  { H }
)  C_  N
188 ssfi 7810 . . . 4  |-  ( ( N  e.  Fin  /\  ( N  \  { H } )  C_  N
)  ->  ( N  \  { H } )  e.  Fin )
18947, 187, 188sylancl 675 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( N  \  { H } )  e.  Fin )
1906, 12, 18, 24, 39, 186, 189findcard2d 7831 . 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 511 . . . . . . . 8  |-  ( k  =  H  ->  (
l  =  I  <->  ( l  =  I  /\  k  =  H ) ) )
192191ifbid 3894 . . . . . . 7  |-  ( k  =  H  ->  if ( l  =  I ,  .1.  ,  .0.  )  =  if (
( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
193 iftrue 3878 . . . . . . 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 3878 . . . . . . . 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 401 . . . . . . 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 2515 . . . . . 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 473 . . . . 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 3881 . . . . . . 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 473 . . . . . 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 22 . . . . . . . . . . 11  |-  ( -.  k  =  H  ->  -.  k  =  H
)
201200neqned 2650 . . . . . . . . . 10  |-  ( -.  k  =  H  -> 
k  =/=  H )
202201anim2i 579 . . . . . . . . 9  |-  ( ( k  e.  N  /\  -.  k  =  H
)  ->  ( k  e.  N  /\  k  =/=  H ) )
203202adantlr 729 . . . . . . . 8  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  (
k  e.  N  /\  k  =/=  H ) )
204 eldifsn 4088 . . . . . . . 8  |-  ( k  e.  ( N  \  { H } )  <->  ( k  e.  N  /\  k  =/=  H ) )
205203, 204sylibr 217 . . . . . . 7  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  k  e.  ( N  \  { H } ) )
206205iftrued 3880 . . . . . 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 412 . . . . . . . 8  |-  ( -.  k  =  H  -> 
( l  =  I  <-> 
( k  =  H  \/  l  =  I ) ) )
208200intnand 930 . . . . . . . . . 10  |-  ( -.  k  =  H  ->  -.  ( l  =  I  /\  k  =  H ) )
209208iffalsed 3883 . . . . . . . . 9  |-  ( -.  k  =  H  ->  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )  =  .0.  )
210209eqcomd 2477 . . . . . . . 8  |-  ( -.  k  =  H  ->  .0.  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )
)
211207, 210ifbieq1d 3895 . . . . . . 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 473 . . . . . 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 2509 . . . . 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 808 . . . 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 6375 . . 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 5882 . 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 2521 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 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390   (/)c0 3722   ifcif 3872   {csn 3959    X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    ^m cmap 7490   Fincfn 7587   Basecbs 15199   +g cplusg 15268   .rcmulr 15269   0gc0g 15416   Mndcmnd 16613   1rcur 17813   Ringcrg 17858   CRingccrg 17859   Mat cmat 19509   maDet cmdat 19686   maAdju cmadu 19734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-xor 1431  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-reverse 12717  df-s2 13003  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-subg 16892  df-ghm 16959  df-gim 17001  df-cntz 17049  df-oppg 17075  df-symg 17097  df-pmtr 17161  df-psgn 17210  df-evpm 17211  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-subrg 18084  df-sra 18473  df-rgmod 18474  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-dsmm 19372  df-frlm 19387  df-mat 19510  df-mdet 19687  df-madu 19736
This theorem is referenced by:  madutpos  19744
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