MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  m1detdiag Structured version   Unicode version

Theorem m1detdiag 18972
Description: The determinant of a 1-dimensional matrix equals its (single) entry. (Contributed by AV, 6-Aug-2019.)
Hypotheses
Ref Expression
mdetdiag.d  |-  D  =  ( N maDet  R )
mdetdiag.a  |-  A  =  ( N Mat  R )
mdetdiag.b  |-  B  =  ( Base `  A
)
Assertion
Ref Expression
m1detdiag  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( D `  M )  =  ( I M I ) )

Proof of Theorem m1detdiag
Dummy variables  b  p  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetdiag.d . . . 4  |-  D  =  ( N maDet  R )
2 mdetdiag.a . . . 4  |-  A  =  ( N Mat  R )
3 mdetdiag.b . . . 4  |-  B  =  ( Base `  A
)
4 eqid 2443 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2443 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2443 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 eqid 2443 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2443 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib 18962 . . 3  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
1093ad2ant3 1020 . 2  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
11 fveq2 5856 . . . . . . . . 9  |-  ( N  =  { I }  ->  ( SymGrp `  N )  =  ( SymGrp `  {
I } ) )
1211fveq2d 5860 . . . . . . . 8  |-  ( N  =  { I }  ->  ( Base `  ( SymGrp `
 N ) )  =  ( Base `  ( SymGrp `
 { I }
) ) )
1312adantr 465 . . . . . . 7  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  { I } ) ) )
14133ad2ant2 1019 . . . . . 6  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  (
Base `  ( SymGrp `  { I } ) ) )
15 simp2r 1024 . . . . . . 7  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  I  e.  V )
16 eqid 2443 . . . . . . . 8  |-  ( SymGrp `  { I } )  =  ( SymGrp `  {
I } )
17 eqid 2443 . . . . . . . 8  |-  ( Base `  ( SymGrp `  { I } ) )  =  ( Base `  ( SymGrp `
 { I }
) )
18 eqid 2443 . . . . . . . 8  |-  { I }  =  { I }
1916, 17, 18symg1bas 16295 . . . . . . 7  |-  ( I  e.  V  ->  ( Base `  ( SymGrp `  {
I } ) )  =  { { <. I ,  I >. } }
)
2015, 19syl 16 . . . . . 6  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  {
I } ) )  =  { { <. I ,  I >. } }
)
2114, 20eqtrd 2484 . . . . 5  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  { { <. I ,  I >. } } )
2221mpteq1d 4518 . . . 4  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
p  e.  ( Base `  ( SymGrp `  N )
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) )  =  ( p  e. 
{ { <. I ,  I >. } }  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) ) )
23 snex 4678 . . . . . 6  |-  { <. I ,  I >. }  e.  _V
2423a1i 11 . . . . 5  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { <. I ,  I >. }  e.  _V )
25 ovex 6309 . . . . 5  |-  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) )  e. 
_V
26 fveq2 5856 . . . . . . . 8  |-  ( p  =  { <. I ,  I >. }  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) )
27 fveq1 5855 . . . . . . . . . . 11  |-  ( p  =  { <. I ,  I >. }  ->  (
p `  x )  =  ( { <. I ,  I >. } `  x ) )
2827oveq1d 6296 . . . . . . . . . 10  |-  ( p  =  { <. I ,  I >. }  ->  (
( p `  x
) M x )  =  ( ( {
<. I ,  I >. } `
 x ) M x ) )
2928mpteq2dv 4524 . . . . . . . . 9  |-  ( p  =  { <. I ,  I >. }  ->  (
x  e.  N  |->  ( ( p `  x
) M x ) )  =  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) )
3029oveq2d 6297 . . . . . . . 8  |-  ( p  =  { <. I ,  I >. }  ->  (
(mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) )  =  ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) )
3126, 30oveq12d 6299 . . . . . . 7  |-  ( p  =  { <. I ,  I >. }  ->  (
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) )  =  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  { <. I ,  I >. } ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) ) ) )
3231fmptsng 6077 . . . . . 6  |-  ( ( { <. I ,  I >. }  e.  _V  /\  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  { <. I ,  I >. } ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) ) )  e.  _V )  ->  { <. { <. I ,  I >. } ,  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) ) >. }  =  ( p  e.  { { <. I ,  I >. } }  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) ) )
3332eqcomd 2451 . . . . 5  |-  ( ( { <. I ,  I >. }  e.  _V  /\  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  { <. I ,  I >. } ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) ) )  e.  _V )  -> 
( p  e.  { { <. I ,  I >. } }  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) )  =  { <. { <. I ,  I >. } , 
( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  { <. I ,  I >. } ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) ) )
>. } )
3424, 25, 33sylancl 662 . . . 4  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
p  e.  { { <. I ,  I >. } }  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) )  =  { <. { <. I ,  I >. } , 
( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  { <. I ,  I >. } ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) ) )
>. } )
35 eqid 2443 . . . . . . . . . . . . 13  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
36 eqid 2443 . . . . . . . . . . . . 13  |-  { b  e.  ( Base `  ( SymGrp `
 N ) )  |  dom  ( b 
\  _I  )  e. 
Fin }  =  {
b  e.  ( Base `  ( SymGrp `  N )
)  |  dom  (
b  \  _I  )  e.  Fin }
3735, 4, 36, 6psgnfn 16400 . . . . . . . . . . . 12  |-  (pmSgn `  N )  Fn  {
b  e.  ( Base `  ( SymGrp `  N )
)  |  dom  (
b  \  _I  )  e.  Fin }
3819adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  { I } ) )  =  { { <. I ,  I >. } } )
3913, 38eqtrd 2484 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  N )
)  =  { { <. I ,  I >. } } )
40393ad2ant2 1019 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  { { <. I ,  I >. } } )
41 rabeq 3089 . . . . . . . . . . . . . . 15  |-  ( (
Base `  ( SymGrp `  N ) )  =  { { <. I ,  I >. } }  ->  { b  e.  ( Base `  ( SymGrp `  N )
)  |  dom  (
b  \  _I  )  e.  Fin }  =  {
b  e.  { { <. I ,  I >. } }  |  dom  (
b  \  _I  )  e.  Fin } )
4240, 41syl 16 . . . . . . . . . . . . . 14  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { b  e.  ( Base `  ( SymGrp `
 N ) )  |  dom  ( b 
\  _I  )  e. 
Fin }  =  {
b  e.  { { <. I ,  I >. } }  |  dom  (
b  \  _I  )  e.  Fin } )
43 difeq1 3600 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  { <. I ,  I >. }  ->  (
b  \  _I  )  =  ( { <. I ,  I >. }  \  _I  ) )
4443dmeqd 5195 . . . . . . . . . . . . . . . . 17  |-  ( b  =  { <. I ,  I >. }  ->  dom  ( b  \  _I  )  =  dom  ( {
<. I ,  I >. } 
\  _I  ) )
4544eleq1d 2512 . . . . . . . . . . . . . . . 16  |-  ( b  =  { <. I ,  I >. }  ->  ( dom  ( b  \  _I  )  e.  Fin  <->  dom  ( {
<. I ,  I >. } 
\  _I  )  e. 
Fin ) )
4645rabsnif 4084 . . . . . . . . . . . . . . 15  |-  { b  e.  { { <. I ,  I >. } }  |  dom  ( b  \  _I  )  e.  Fin }  =  if ( dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin ,  { { <. I ,  I >. } } ,  (/) )
4746a1i 11 . . . . . . . . . . . . . 14  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { b  e.  { { <. I ,  I >. } }  |  dom  ( b  \  _I  )  e.  Fin }  =  if ( dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin ,  { { <. I ,  I >. } } ,  (/) ) )
48 restidsing 5320 . . . . . . . . . . . . . . . . . . . 20  |-  (  _I  |`  { I } )  =  ( { I }  X.  { I }
)
49 xpsng 6057 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  I  e.  V )  ->  ( { I }  X.  { I } )  =  { <. I ,  I >. } )
5049anidms 645 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  V  ->  ( { I }  X.  { I } )  =  { <. I ,  I >. } )
5148, 50syl5req 2497 . . . . . . . . . . . . . . . . . . 19  |-  ( I  e.  V  ->  { <. I ,  I >. }  =  (  _I  |`  { I } ) )
52 fnsng 5625 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  I  e.  V )  ->  { <. I ,  I >. }  Fn  { I } )
5352anidms 645 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  V  ->  { <. I ,  I >. }  Fn  { I } )
54 fnnfpeq0 6087 . . . . . . . . . . . . . . . . . . . 20  |-  ( {
<. I ,  I >. }  Fn  { I }  ->  ( dom  ( {
<. I ,  I >. } 
\  _I  )  =  (/) 
<->  { <. I ,  I >. }  =  (  _I  |`  { I } ) ) )
5553, 54syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( I  e.  V  ->  ( dom  ( { <. I ,  I >. }  \  _I  )  =  (/)  <->  { <. I ,  I >. }  =  (  _I  |`  { I } ) ) )
5651, 55mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( I  e.  V  ->  dom  ( { <. I ,  I >. }  \  _I  )  =  (/) )
57 0fin 7749 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  Fin
5856, 57syl6eqel 2539 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin )
5958adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  dom  ( {
<. I ,  I >. } 
\  _I  )  e. 
Fin )
60593ad2ant2 1019 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin )
6160iftrued 3934 . . . . . . . . . . . . . 14  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  if ( dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin ,  { { <. I ,  I >. } } ,  (/) )  =  { { <. I ,  I >. } } )
6242, 47, 613eqtrrd 2489 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { { <. I ,  I >. } }  =  { b  e.  ( Base `  ( SymGrp `
 N ) )  |  dom  ( b 
\  _I  )  e. 
Fin } )
6362fneq2d 5662 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N )  Fn  { { <. I ,  I >. } }  <->  (pmSgn `  N
)  Fn  { b  e.  ( Base `  ( SymGrp `
 N ) )  |  dom  ( b 
\  _I  )  e. 
Fin } ) )
6437, 63mpbiri 233 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (pmSgn `  N )  Fn  { { <. I ,  I >. } } )
6523snid 4042 . . . . . . . . . . 11  |-  { <. I ,  I >. }  e.  { { <. I ,  I >. } }
66 fvco2 5933 . . . . . . . . . . 11  |-  ( ( (pmSgn `  N )  Fn  { { <. I ,  I >. } }  /\  {
<. I ,  I >. }  e.  { { <. I ,  I >. } }
)  ->  ( (
( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } )  =  ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. I ,  I >. } ) ) )
6764, 65, 66sylancl 662 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } )  =  ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. I ,  I >. } ) ) )
68 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( N  =  { I }  ->  (pmSgn `  N )  =  (pmSgn `  { I } ) )
6968adantr 465 . . . . . . . . . . . . . 14  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  (pmSgn `  N
)  =  (pmSgn `  { I } ) )
70693ad2ant2 1019 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (pmSgn `  N )  =  (pmSgn `  { I } ) )
7170fveq1d 5858 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N ) `  { <. I ,  I >. } )  =  ( (pmSgn `  { I } ) `  { <. I ,  I >. } ) )
72 snidg 4040 . . . . . . . . . . . . . . . . . 18  |-  ( {
<. I ,  I >. }  e.  _V  ->  { <. I ,  I >. }  e.  { { <. I ,  I >. } } )
7323, 72mp1i 12 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  { <. I ,  I >. }  e.  { { <. I ,  I >. } } )
7473, 19eleqtrrd 2534 . . . . . . . . . . . . . . . 16  |-  ( I  e.  V  ->  { <. I ,  I >. }  e.  ( Base `  ( SymGrp `  { I } ) ) )
7574ancli 551 . . . . . . . . . . . . . . 15  |-  ( I  e.  V  ->  (
I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) ) )
7675adantl 466 . . . . . . . . . . . . . 14  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( I  e.  V  /\  { <. I ,  I >. }  e.  ( Base `  ( SymGrp `  { I } ) ) ) )
77763ad2ant2 1019 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) ) )
78 eqid 2443 . . . . . . . . . . . . . 14  |-  (pmSgn `  { I } )  =  (pmSgn `  {
I } )
7918, 16, 17, 78psgnsn 16419 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) )  -> 
( (pmSgn `  {
I } ) `  { <. I ,  I >. } )  =  1 )
8077, 79syl 16 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  { I } ) `  { <. I ,  I >. } )  =  1 )
8171, 80eqtrd 2484 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N ) `  { <. I ,  I >. } )  =  1 )
8281fveq2d 5860 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. I ,  I >. } ) )  =  ( ( ZRHom `  R
) `  1 )
)
83 crngring 17083 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  R  e.  Ring )
84833ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  R  e.  Ring )
85 eqid 2443 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
865, 85zrh1 18423 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( ( ZRHom `  R ) `  1 )  =  ( 1r `  R
) )
8784, 86syl 16 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  1 )  =  ( 1r `  R ) )
8867, 82, 873eqtrd 2488 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } )  =  ( 1r `  R ) )
89 simp2l 1023 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  N  =  { I } )
9089mpteq1d 4518 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) )  =  ( x  e.  { I }  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) )
9190oveq2d 6297 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) )  =  ( (mulGrp `  R )  gsumg  ( x  e.  { I }  |->  ( ( {
<. I ,  I >. } `
 x ) M x ) ) ) )
928ringmgp 17078 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
9383, 92syl 16 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e.  Mnd )
94933ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (mulGrp `  R )  e.  Mnd )
95 snidg 4040 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  I  e.  { I } )
9695adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  I  e.  { I } )
97 eleq2 2516 . . . . . . . . . . . . . . . . 17  |-  ( N  =  { I }  ->  ( I  e.  N  <->  I  e.  { I }
) )
9897adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( I  e.  N  <->  I  e.  { I } ) )
9996, 98mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  I  e.  N )
1003eleq2i 2521 . . . . . . . . . . . . . . . 16  |-  ( M  e.  B  <->  M  e.  ( Base `  A )
)
101100biimpi 194 . . . . . . . . . . . . . . 15  |-  ( M  e.  B  ->  M  e.  ( Base `  A
) )
102 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  N  /\  M  e.  ( Base `  A ) )  ->  I  e.  N )
103 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  N  /\  M  e.  ( Base `  A ) )  ->  M  e.  ( Base `  A ) )
104102, 102, 1033jca 1177 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
10599, 101, 104syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( ( N  =  {
I }  /\  I  e.  V )  /\  M  e.  B )  ->  (
I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
1061053adant1 1015 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
107 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  R )  =  (
Base `  R )
1082, 107matecl 18800 . . . . . . . . . . . . 13  |-  ( ( I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( I M I )  e.  ( Base `  R ) )
109106, 108syl 16 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  R
) )
1108, 107mgpbas 17021 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
111109, 110syl6eleq 2541 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  (mulGrp `  R ) ) )
112 eqid 2443 . . . . . . . . . . . 12  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
113 fveq2 5856 . . . . . . . . . . . . . 14  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  x )  =  ( { <. I ,  I >. } `  I ) )
114 eqvisset 3103 . . . . . . . . . . . . . . 15  |-  ( x  =  I  ->  I  e.  _V )
115 fvsng 6090 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  _V  /\  I  e.  _V )  ->  ( { <. I ,  I >. } `  I
)  =  I )
116114, 114, 115syl2anc 661 . . . . . . . . . . . . . 14  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  I )  =  I )
117113, 116eqtrd 2484 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  x )  =  I )
118 id 22 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  x  =  I )
119117, 118oveq12d 6299 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
( { <. I ,  I >. } `  x
) M x )  =  ( I M I ) )
120112, 119gsumsn 16855 . . . . . . . . . . 11  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  I  e.  V  /\  ( I M I )  e.  ( Base `  (mulGrp `  R ) ) )  ->  ( (mulGrp `  R )  gsumg  ( x  e.  {
I }  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) )  =  ( I M I ) )
12194, 15, 111, 120syl3anc 1229 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( x  e.  { I }  |->  ( ( {
<. I ,  I >. } `
 x ) M x ) ) )  =  ( I M I ) )
12291, 121eqtrd 2484 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) )  =  ( I M I ) )
12388, 122oveq12d 6299 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) )  =  ( ( 1r `  R ) ( .r
`  R ) ( I M I ) ) )
124993ad2ant2 1019 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  I  e.  N )
1251013ad2ant3 1020 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  M  e.  ( Base `  A
) )
126124, 124, 125, 108syl3anc 1229 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  R
) )
127107, 7, 85ringlidm 17096 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
I M I )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( I M I ) )  =  ( I M I ) )
12884, 126, 127syl2anc 661 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( I M I ) )  =  ( I M I ) )
129123, 128eqtrd 2484 . . . . . . 7  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) )  =  ( I M I ) )
130129opeq2d 4209 . . . . . 6  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  <. { <. I ,  I >. } , 
( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  { <. I ,  I >. } ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) ) )
>.  =  <. { <. I ,  I >. } , 
( I M I ) >. )
131130sneqd 4026 . . . . 5  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { <. {
<. I ,  I >. } ,  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) ) >. }  =  { <. { <. I ,  I >. } , 
( I M I ) >. } )
132 ovex 6309 . . . . . 6  |-  ( I M I )  e. 
_V
133 eqidd 2444 . . . . . . 7  |-  ( y  =  { <. I ,  I >. }  ->  (
I M I )  =  ( I M I ) )
134133fmptsng 6077 . . . . . 6  |-  ( ( { <. I ,  I >. }  e.  _V  /\  ( I M I )  e.  _V )  ->  { <. { <. I ,  I >. } ,  ( I M I )
>. }  =  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )
13524, 132, 134sylancl 662 . . . . 5  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { <. {
<. I ,  I >. } ,  ( I M I ) >. }  =  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )
136131, 135eqtrd 2484 . . . 4  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { <. {
<. I ,  I >. } ,  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x
) M x ) ) ) ) >. }  =  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )
13722, 34, 1363eqtrd 2488 . . 3  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
p  e.  ( Base `  ( SymGrp `  N )
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) )  =  ( y  e. 
{ { <. I ,  I >. } }  |->  ( I M I ) ) )
138137oveq2d 6297 . 2  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( R  gsumg  ( p  e.  (
Base `  ( SymGrp `  N ) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( x  e.  N  |->  ( ( p `  x
) M x ) ) ) ) ) )  =  ( R 
gsumg  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) ) )
139 ringmnd 17081 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
14083, 139syl 16 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Mnd )
1411403ad2ant1 1018 . . 3  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  R  e.  Mnd )
142107, 133gsumsn 16855 . . 3  |-  ( ( R  e.  Mnd  /\  {
<. I ,  I >. }  e.  _V  /\  (
I M I )  e.  ( Base `  R
) )  ->  ( R  gsumg  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )  =  ( I M I ) )
143141, 24, 126, 142syl3anc 1229 . 2  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( R  gsumg  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )  =  ( I M I ) )
14410, 138, 1433eqtrd 2488 1  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( D `  M )  =  ( I M I ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   {crab 2797   _Vcvv 3095    \ cdif 3458   (/)c0 3770   ifcif 3926   {csn 4014   <.cop 4020    |-> cmpt 4495    _I cid 4780    X. cxp 4987   dom cdm 4989    |` cres 4991    o. ccom 4993    Fn wfn 5573   ` cfv 5578  (class class class)co 6281   Fincfn 7518   1c1 9496   Basecbs 14509   .rcmulr 14575    gsumg cgsu 14715   Mndcmnd 15793   SymGrpcsymg 16276  pmSgncpsgn 16388  mulGrpcmgp 17015   1rcur 17027   Ringcrg 17072   CRingccrg 17073   ZRHomczrh 18410   Mat cmat 18782   maDet cmdat 18959
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-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-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-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 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-rp 11230  df-fz 11682  df-fzo 11804  df-seq 12087  df-exp 12146  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-substr 12525  df-splice 12526  df-reverse 12527  df-s2 12792  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-0g 14716  df-gsum 14717  df-prds 14722  df-pws 14724  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-mulg 15934  df-subg 16072  df-ghm 16139  df-gim 16181  df-cntz 16229  df-oppg 16255  df-symg 16277  df-pmtr 16341  df-psgn 16390  df-cmn 16674  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-rnghom 17238  df-subrg 17301  df-sra 17692  df-rgmod 17693  df-cnfld 18295  df-zring 18363  df-zrh 18414  df-dsmm 18636  df-frlm 18651  df-mat 18783  df-mdet 18960
This theorem is referenced by:  chpmat1d  19210
  Copyright terms: Public domain W3C validator