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Theorem maducoeval2 19119
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 2516 . . . . . . . 8  |-  ( m  =  (/)  ->  ( k  e.  m  <->  k  e.  (/) ) )
21ifbid 3948 . . . . . . 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 3945 . . . . . 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 6348 . . . . 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 5860 . . . 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 2457 . . 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 2516 . . . . . . . 8  |-  ( m  =  n  ->  (
k  e.  m  <->  k  e.  n ) )
87ifbid 3948 . . . . . . 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 3945 . . . . . 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 6348 . . . . 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 5860 . . . 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 2457 . . 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 2516 . . . . . . . 8  |-  ( m  =  ( n  u. 
{ r } )  ->  ( k  e.  m  <->  k  e.  ( n  u.  { r } ) ) )
1413ifbid 3948 . . . . . . 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 3945 . . . . . 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 6348 . . . . 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 5860 . . . 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 2457 . . 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 2516 . . . . . . . 8  |-  ( m  =  ( N  \  { H } )  -> 
( k  e.  m  <->  k  e.  ( N  \  { H } ) ) )
2019ifbid 3948 . . . . . . 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 3945 . . . . . 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 6348 . . . . 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 5860 . . . 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 2457 . . 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 19118 . . . . 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 1221 . . . 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 3774 . . . . . . . 8  |-  -.  k  e.  (/)
34 iffalse 3935 . . . . . . . 8  |-  ( -.  k  e.  (/)  ->  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
3533, 34mp1i 12 . . . . . . 7  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( k  e.  (/) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) )  =  ( k M l ) )
3635ifeq2d 3945 . . . . . 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 6347 . . . . 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 5859 . . . 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 2502 . . 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 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
41 eqid 2443 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
42 eqid 2443 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
43 simpl1l 1048 . . . . . . 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 1022 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  M  e.  B )
4525, 28matrcl 18891 . . . . . . . . . 10  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
4645simpld 459 . . . . . . . . 9  |-  ( M  e.  B  ->  N  e.  Fin )
4744, 46syl 16 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  N  e.  Fin )
4847adantr 465 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  N  e.  Fin )
49 simp1l 1021 . . . . . . . . . . 11  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  R  e.  CRing
)
5049ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  R  e.  CRing )
51 crngring 17187 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  R  e.  Ring )
5250, 51syl 16 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  R  e.  Ring )
5340, 30ring0cl 17198 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .0.  e.  ( Base `  R )
)
5452, 53syl 16 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  .0.  e.  ( Base `  R ) )
55 simpl1r 1049 . . . . . . . . . . 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 18901 . . . . . . . . . . 11  |-  ( M  e.  B  ->  M  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
57 elmapi 7442 . . . . . . . . . . 11  |-  ( M  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
5855, 56, 573syl 20 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  M : ( N  X.  N ) --> ( Base `  R
) )
5958adantr 465 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  M : ( N  X.  N ) --> (
Base `  R )
)
60 eldifi 3611 . . . . . . . . . . . 12  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  r  e.  ( N 
\  { H }
) )
6160ad2antll 728 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  r  e.  ( N  \  { H } ) )
6261eldifad 3473 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  r  e.  N
)
6362adantr 465 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  r  e.  N )
64 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  l  e.  N )
6559, 63, 64fovrnd 6432 . . . . . . . 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 3969 . . . . . . 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 17197 . . . . . . . . 9  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
6852, 67syl 16 . . . . . . . 8  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  .1.  e.  ( Base `  R ) )
6968, 54ifcld 3969 . . . . . . 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 1016 . . . . . . . . 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 6431 . . . . . . . . . 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 1193 . . . . . . . . 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 3969 . . . . . . . 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 3969 . . . . . . 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 1001 . . . . . . . 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 6432 . . . . . . 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 1002 . . . . . . 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 4141 . . . . . . . 8  |-  ( r  e.  ( N  \  { H } )  -> 
r  =/=  H )
7961, 78syl 16 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  r  =/=  H
)
8026, 40, 41, 42, 43, 48, 66, 69, 74, 76, 62, 77, 79mdetero 19089 . . . . . 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 3971 . . . . . . . . . . . . . . . . 17  |-  if ( -.  l  =  I ,  ( r M l ) ,  .0.  )  =  if (
l  =  I ,  .0.  ,  ( r M l ) )
8281eqcomi 2456 . . . . . . . . . . . . . . . 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 6365 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( r M I )  e.  ( Base `  R ) )
8640, 42, 29ringridm 17201 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .1.  )  =  ( r M I ) )
8752, 85, 86syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( ( r M I ) ( .r
`  R )  .1.  )  =  ( r M I ) )
8887adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  /\  l  =  I
)  ->  ( (
r M I ) ( .r `  R
)  .1.  )  =  ( r M I ) )
89 oveq2 6289 . . . . . . . . . . . . . . . . . . . 20  |-  ( l  =  I  ->  (
r M l )  =  ( r M I ) )
9089adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  /\  l  =  I
)  ->  ( r M l )  =  ( r M I ) )
9188, 90eqtr4d 2487 . . . . . . . . . . . . . . . . . 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 3956 . . . . . . . . . . . . . . . . 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 17214 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  Ring  /\  (
r M I )  e.  ( Base `  R
) )  ->  (
( r M I ) ( .r `  R )  .0.  )  =  .0.  )
9452, 85, 93syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  ( ( r M I ) ( .r
`  R )  .0.  )  =  .0.  )
9594ifeq2d 3945 . . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . 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 2496 . . . . . . . . . . . . . . 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 6299 . . . . . . . . . . . . . 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 17185 . . . . . . . . . . . . . . . 16  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
10052, 99syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  R  e.  Mnd )
101 id 22 . . . . . . . . . . . . . . . . 17  |-  ( -.  l  =  I  ->  -.  l  =  I
)
102 imnan 422 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  l  =  I  ->  -.  l  =  I )  <->  -.  ( -.  l  =  I  /\  l  =  I
) )
103101, 102mpbi 208 . . . . . . . . . . . . . . . 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 19115 . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . 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 pm2.1 417 . . . . . . . . . . . . . . 15  |-  ( -.  l  =  I  \/  l  =  I )
108 iftrue 3932 . . . . . . . . . . . . . . 15  |-  ( ( -.  l  =  I  \/  l  =  I )  ->  if (
( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( r M l ) )
109107, 108mp1i 12 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  l  e.  N )  ->  if ( ( -.  l  =  I  \/  l  =  I ) ,  ( r M l ) ,  .0.  )  =  ( r M l ) )
11098, 106, 1093eqtr2d 2490 . . . . . . . . . . . . 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 1016 . . . . . . . . . . . 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 6288 . . . . . . . . . . . . 13  |-  ( k  =  r  ->  (
k M l )  =  ( r M l ) )
113112eqeq2d 2457 . . . . . . . . . . . 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 222 . . . . . . . . . . 11  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k  =  r  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( k M l ) ) )
115114imp 429 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R
) ( ( r M I ) ( .r `  R ) if ( l  =  I ,  .1.  ,  .0.  ) ) )  =  ( k M l ) )
116 iftrue 3932 . . . . . . . . . . 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 466 . . . . . . . . . 10  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  if (
k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) )
11879neneqd 2645 . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . 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 2447 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  =  H  <->  r  =  H ) )
121120notbid 294 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  =  H  <->  -.  r  =  H ) )
122119, 121syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k  =  r  ->  -.  k  =  H ) )
123122imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  -.  k  =  H )
124123iffalsed 3937 . . . . . . . . . . 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 3612 . . . . . . . . . . . . . . . 16  |-  ( r  e.  ( ( N 
\  { H }
)  \  n )  ->  -.  r  e.  n
)
126125ad2antll 728 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  -.  r  e.  n )
1271263ad2ant1 1018 . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  (
k  e.  n  <->  r  e.  n ) )
129128notbid 294 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  ( -.  k  e.  n  <->  -.  r  e.  n ) )
130127, 129syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  ->  ( k  =  r  ->  -.  k  e.  n ) )
131130imp 429 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  k  =  r
)  ->  -.  k  e.  n )
132131iffalsed 3937 . . . . . . . . . . 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 2484 . . . . . . . . . 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 2494 . . . . . . . . 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 3935 . . . . . . . . . 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 466 . . . . . . . . 9  |-  ( ( ( ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N )  /\  (
n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n
) ) )  /\  k  e.  N  /\  l  e.  N )  /\  -.  k  =  r )  ->  if (
k  =  r ,  ( if ( l  =  I ,  .0.  ,  ( r M l ) ) ( +g  `  R ) ( ( r M I ) ( .r `  R
) if ( l  =  I ,  .1.  ,  .0.  ) ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
137134, 136pm2.61dan 791 . . . . . . . 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 6346 . . . . . . 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 5860 . . . . . 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 2726 . . . . . . . . . . . . . . 15  |-  ( k  =  H  ->  (
r  =/=  k  <->  r  =/=  H ) )
141140biimparc 487 . . . . . . . . . . . . . 14  |-  ( ( r  =/=  H  /\  k  =  H )  ->  r  =/=  k )
142141necomd 2714 . . . . . . . . . . . . 13  |-  ( ( r  =/=  H  /\  k  =  H )  ->  k  =/=  r )
143142neneqd 2645 . . . . . . . . . . . 12  |-  ( ( r  =/=  H  /\  k  =  H )  ->  -.  k  =  r )
144143iffalsed 3937 . . . . . . . . . . 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 3932 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
147146ifeq2d 3945 . . . . . . . . . . 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 3932 . . . . . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( n  u.  {
r } ) ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( l  =  I ,  .1.  ,  .0.  ) )
150144, 147, 1493eqtr4d 2494 . . . . . . . . . 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 3945 . . . . . . . . . . . . . 14  |-  ( k  =  r  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( l  =  I ,  .0.  ,  ( r M l ) ) )
152 ssnid 4043 . . . . . . . . . . . . . . . . 17  |-  r  e. 
{ r }
153 elun2 3657 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  { r }  ->  r  e.  ( n  u.  { r } ) )
154152, 153ax-mp 5 . . . . . . . . . . . . . . . 16  |-  r  e.  ( n  u.  {
r } )
155 eleq1 2515 . . . . . . . . . . . . . . . 16  |-  ( k  =  r  ->  (
k  e.  ( n  u.  { r } )  <->  r  e.  ( n  u.  { r } ) ) )
156154, 155mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( k  =  r  ->  k  e.  ( n  u.  {
r } ) )
157156iftrued 3934 . . . . . . . . . . . . . 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 3932 . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  H  /\  -.  k  =  H )  /\  k  =  r )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
161 iffalse 3935 . . . . . . . . . . . . . 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 385 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  n  ->  (
k  e.  n  \/  k  =  r ) )
163 orel2 383 . . . . . . . . . . . . . . . . 17  |-  ( -.  k  =  r  -> 
( ( k  e.  n  \/  k  =  r )  ->  k  e.  n ) )
164162, 163impbid2 204 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  ( k  e.  n  \/  k  =  r ) ) )
165 elun 3630 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( n  u. 
{ r } )  <-> 
( k  e.  n  \/  k  e.  { r } ) )
166 elsn 4028 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  { r }  <-> 
k  =  r )
167166orbi2i 519 . . . . . . . . . . . . . . . . 17  |-  ( ( k  e.  n  \/  k  e.  { r } )  <->  ( k  e.  n  \/  k  =  r ) )
168165, 167bitr2i 250 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  n  \/  k  =  r )  <-> 
k  e.  ( n  u.  { r } ) )
169164, 168syl6bb 261 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  r  -> 
( k  e.  n  <->  k  e.  ( n  u. 
{ r } ) ) )
170169ifbid 3948 . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( ( ( r  =/=  H  /\  -.  k  =  H )  /\  -.  k  =  r )  ->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
173160, 172pm2.61dan 791 . . . . . . . . . . 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 3935 . . . . . . . . . . . . 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 3945 . . . . . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  -.  k  =  H
)  ->  if (
k  =  r ,  if ( l  =  I ,  .0.  , 
( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) )  =  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) )
177 iffalse 3935 . . . . . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( ( r  =/=  H  /\  -.  k  =  H
)  ->  if (
k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( n  u. 
{ r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
179173, 176, 1783eqtr4d 2494 . . . . . . . . . 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 791 . . . . . . . . 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 6348 . . . . . . . 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 5860 . . . . . . 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 16 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  r ,  if ( l  =  I ,  .0.  ,  ( r M l ) ) ,  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) ) )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) )
18480, 139, 1833eqtr3d 2492 . . . . 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 2457 . . . 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 207 . . 3  |-  ( ( ( ( R  e. 
CRing  /\  M  e.  B
)  /\  I  e.  N  /\  H  e.  N
)  /\  ( n  C_  ( N  \  { H } )  /\  r  e.  ( ( N  \  { H } )  \  n ) ) )  ->  ( ( I ( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  n ,  if ( l  =  I ,  .0.  , 
( k M l ) ) ,  ( k M l ) ) ) ) )  ->  ( I ( J `  M ) H )  =  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if (
k  e.  ( n  u.  { r } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) ) ) ) ) )
187 difss 3616 . . . 4  |-  ( N 
\  { H }
)  C_  N
188 ssfi 7742 . . . 4  |-  ( ( N  e.  Fin  /\  ( N  \  { H } )  C_  N
)  ->  ( N  \  { H } )  e.  Fin )
18947, 187, 188sylancl 662 . . 3  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( N  \  { H } )  e.  Fin )
1906, 12, 18, 24, 39, 186, 189findcard2d 7764 . 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 503 . . . . . . . 8  |-  ( k  =  H  ->  (
l  =  I  <->  ( l  =  I  /\  k  =  H ) ) )
192191ifbid 3948 . . . . . . 7  |-  ( k  =  H  ->  if ( l  =  I ,  .1.  ,  .0.  )  =  if (
( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
193 iftrue 3932 . . . . . . 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 3932 . . . . . . . 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 394 . . . . . . 7  |-  ( k  =  H  ->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) )  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) )
196192, 193, 1953eqtr4d 2494 . . . . . 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 466 . . . . 5  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
198 iffalse 3935 . . . . . . 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 466 . . . . . 6  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  if ( k  e.  ( N  \  { H } ) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )  =  if ( k  e.  ( N 
\  { H }
) ,  if ( l  =  I ,  .0.  ,  ( k M l ) ) ,  ( k M l ) ) )
200 id 22 . . . . . . . . . . 11  |-  ( -.  k  =  H  ->  -.  k  =  H
)
201200neqned 2646 . . . . . . . . . 10  |-  ( -.  k  =  H  -> 
k  =/=  H )
202201anim2i 569 . . . . . . . . 9  |-  ( ( k  e.  N  /\  -.  k  =  H
)  ->  ( k  e.  N  /\  k  =/=  H ) )
203202adantlr 714 . . . . . . . 8  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  (
k  e.  N  /\  k  =/=  H ) )
204 eldifsn 4140 . . . . . . . 8  |-  ( k  e.  ( N  \  { H } )  <->  ( k  e.  N  /\  k  =/=  H ) )
205203, 204sylibr 212 . . . . . . 7  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  k  e.  ( N  \  { H } ) )
206205iftrued 3934 . . . . . 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 405 . . . . . . . 8  |-  ( -.  k  =  H  -> 
( l  =  I  <-> 
( k  =  H  \/  l  =  I ) ) )
208200intnand 916 . . . . . . . . . 10  |-  ( -.  k  =  H  ->  -.  ( l  =  I  /\  k  =  H ) )
209208iffalsed 3937 . . . . . . . . 9  |-  ( -.  k  =  H  ->  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )  =  .0.  )
210209eqcomd 2451 . . . . . . . 8  |-  ( -.  k  =  H  ->  .0.  =  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  )
)
211207, 210ifbieq1d 3949 . . . . . . 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 466 . . . . . 6  |-  ( ( ( k  e.  N  /\  l  e.  N
)  /\  -.  k  =  H )  ->  if ( l  =  I ,  .0.  ,  ( k M l ) )  =  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
213199, 206, 2123eqtrd 2488 . . . . 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 791 . . . 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 6347 . . 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 5859 . 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 2500 1  |-  ( ( ( R  e.  CRing  /\  M  e.  B )  /\  I  e.  N  /\  H  e.  N
)  ->  ( I
( J `  M
) H )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( ( k  =  H  \/  l  =  I ) ,  if ( ( l  =  I  /\  k  =  H ) ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   (/)c0 3770   ifcif 3926   {csn 4014    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283    ^m cmap 7422   Fincfn 7518   Basecbs 14613   +g cplusg 14678   .rcmulr 14679   0gc0g 14818   Mndcmnd 15897   1rcur 17131   Ringcrg 17176   CRingccrg 17177   Mat cmat 18886   maDet cmdat 19063   maAdju cmadu 19111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-xor 1365  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-rp 11231  df-fz 11683  df-fzo 11806  df-seq 12089  df-exp 12148  df-hash 12387  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12794  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-hom 14702  df-cco 14703  df-0g 14820  df-gsum 14821  df-prds 14826  df-pws 14828  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-mhm 15944  df-submnd 15945  df-grp 16035  df-minusg 16036  df-mulg 16038  df-subg 16176  df-ghm 16243  df-gim 16285  df-cntz 16333  df-oppg 16359  df-symg 16381  df-pmtr 16445  df-psgn 16494  df-evpm 16495  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-ring 17178  df-cring 17179  df-oppr 17250  df-dvdsr 17268  df-unit 17269  df-invr 17299  df-dvr 17310  df-rnghom 17342  df-drng 17376  df-subrg 17405  df-sra 17796  df-rgmod 17797  df-cnfld 18399  df-zring 18467  df-zrh 18518  df-dsmm 18740  df-frlm 18755  df-mat 18887  df-mdet 19064  df-madu 19113
This theorem is referenced by:  madutpos  19121
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