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Theorem m1detdiag 18859
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 2460 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2460 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2460 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 eqid 2460 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2460 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib 18849 . . 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 1014 . 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 5857 . . . . . . . . 9  |-  ( N  =  { I }  ->  ( SymGrp `  N )  =  ( SymGrp `  {
I } ) )
1211fveq2d 5861 . . . . . . . 8  |-  ( N  =  { I }  ->  ( Base `  ( SymGrp `
 N ) )  =  ( Base `  ( SymGrp `
 { I }
) ) )
1312adantr 465 . . . . . . 7  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  { I } ) ) )
14133ad2ant2 1013 . . . . . 6  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  (
Base `  ( SymGrp `  { I } ) ) )
15 simp2r 1018 . . . . . . 7  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  I  e.  V )
16 eqid 2460 . . . . . . . 8  |-  ( SymGrp `  { I } )  =  ( SymGrp `  {
I } )
17 eqid 2460 . . . . . . . 8  |-  ( Base `  ( SymGrp `  { I } ) )  =  ( Base `  ( SymGrp `
 { I }
) )
18 eqid 2460 . . . . . . . 8  |-  { I }  =  { I }
1916, 17, 18symg1bas 16209 . . . . . . 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 2501 . . . . 5  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  { { <. I ,  I >. } } )
2221mpteq1d 4521 . . . 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 4681 . . . . . 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 6300 . . . . 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 5857 . . . . . . . 8  |-  ( p  =  { <. I ,  I >. }  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) )
27 fveq1 5856 . . . . . . . . . . 11  |-  ( p  =  { <. I ,  I >. }  ->  (
p `  x )  =  ( { <. I ,  I >. } `  x ) )
2827oveq1d 6290 . . . . . . . . . 10  |-  ( p  =  { <. I ,  I >. }  ->  (
( p `  x
) M x )  =  ( ( {
<. I ,  I >. } `
 x ) M x ) )
2928mpteq2dv 4527 . . . . . . . . 9  |-  ( p  =  { <. I ,  I >. }  ->  (
x  e.  N  |->  ( ( p `  x
) M x ) )  =  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) )
3029oveq2d 6291 . . . . . . . 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 6293 . . . . . . 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 6073 . . . . . 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 2468 . . . . 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 2460 . . . . . . . . . . . . 13  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
36 eqid 2460 . . . . . . . . . . . . 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 16315 . . . . . . . . . . . 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 2501 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  N )
)  =  { { <. I ,  I >. } } )
40393ad2ant2 1013 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  { { <. I ,  I >. } } )
41 rabeq 3100 . . . . . . . . . . . . . . 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 3608 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  { <. I ,  I >. }  ->  (
b  \  _I  )  =  ( { <. I ,  I >. }  \  _I  ) )
4443dmeqd 5196 . . . . . . . . . . . . . . . . 17  |-  ( b  =  { <. I ,  I >. }  ->  dom  ( b  \  _I  )  =  dom  ( {
<. I ,  I >. } 
\  _I  ) )
4544eleq1d 2529 . . . . . . . . . . . . . . . 16  |-  ( b  =  { <. I ,  I >. }  ->  ( dom  ( b  \  _I  )  e.  Fin  <->  dom  ( {
<. I ,  I >. } 
\  _I  )  e. 
Fin ) )
4645rabsnif 4089 . . . . . . . . . . . . . . 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 5321 . . . . . . . . . . . . . . . . . . . 20  |-  (  _I  |`  { I } )  =  ( { I }  X.  { I }
)
49 xpsng 6053 . . . . . . . . . . . . . . . . . . . . 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 2514 . . . . . . . . . . . . . . . . . . 19  |-  ( I  e.  V  ->  { <. I ,  I >. }  =  (  _I  |`  { I } ) )
52 fnsng 5626 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  I  e.  V )  ->  { <. I ,  I >. }  Fn  { I } )
5352anidms 645 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  V  ->  { <. I ,  I >. }  Fn  { I } )
54 fnnfpeq0 6083 . . . . . . . . . . . . . . . . . . . 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 7737 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  Fin
5856, 57syl6eqel 2556 . . . . . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin )
61 iftrue 3938 . . . . . . . . . . . . . . 15  |-  ( dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin  ->  if ( dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin ,  { { <. I ,  I >. } } ,  (/) )  =  { { <. I ,  I >. } } )
6260, 61syl 16 . . . . . . . . . . . . . 14  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  if ( dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin ,  { { <. I ,  I >. } } ,  (/) )  =  { { <. I ,  I >. } } )
6342, 47, 623eqtrrd 2506 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  { { <. I ,  I >. } }  =  { b  e.  ( Base `  ( SymGrp `
 N ) )  |  dom  ( b 
\  _I  )  e. 
Fin } )
6463fneq2d 5663 . . . . . . . . . . . 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 } ) )
6537, 64mpbiri 233 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (pmSgn `  N )  Fn  { { <. I ,  I >. } } )
6623snid 4048 . . . . . . . . . . 11  |-  { <. I ,  I >. }  e.  { { <. I ,  I >. } }
67 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 >. } ) ) )
6865, 66, 67sylancl 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 >. } ) ) )
69 fveq2 5857 . . . . . . . . . . . . . . 15  |-  ( N  =  { I }  ->  (pmSgn `  N )  =  (pmSgn `  { I } ) )
7069adantr 465 . . . . . . . . . . . . . 14  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  (pmSgn `  N
)  =  (pmSgn `  { I } ) )
71703ad2ant2 1013 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (pmSgn `  N )  =  (pmSgn `  { I } ) )
7271fveq1d 5859 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N ) `  { <. I ,  I >. } )  =  ( (pmSgn `  { I } ) `  { <. I ,  I >. } ) )
73 snidg 4046 . . . . . . . . . . . . . . . . . 18  |-  ( {
<. I ,  I >. }  e.  _V  ->  { <. I ,  I >. }  e.  { { <. I ,  I >. } } )
7423, 73mp1i 12 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  { <. I ,  I >. }  e.  { { <. I ,  I >. } } )
7574, 19eleqtrrd 2551 . . . . . . . . . . . . . . . 16  |-  ( I  e.  V  ->  { <. I ,  I >. }  e.  ( Base `  ( SymGrp `  { I } ) ) )
7675ancli 551 . . . . . . . . . . . . . . 15  |-  ( I  e.  V  ->  (
I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) ) )
7776adantl 466 . . . . . . . . . . . . . 14  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( I  e.  V  /\  { <. I ,  I >. }  e.  ( Base `  ( SymGrp `  { I } ) ) ) )
78773ad2ant2 1013 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) ) )
79 eqid 2460 . . . . . . . . . . . . . 14  |-  (pmSgn `  { I } )  =  (pmSgn `  {
I } )
8018, 16, 17, 79psgnsn 16334 . . . . . . . . . . . . 13  |-  ( ( I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) )  -> 
( (pmSgn `  {
I } ) `  { <. I ,  I >. } )  =  1 )
8178, 80syl 16 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  { I } ) `  { <. I ,  I >. } )  =  1 )
8272, 81eqtrd 2501 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N ) `  { <. I ,  I >. } )  =  1 )
8382fveq2d 5861 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. I ,  I >. } ) )  =  ( ( ZRHom `  R
) `  1 )
)
84 crngrng 16989 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  R  e.  Ring )
85843ad2ant1 1012 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  R  e.  Ring )
86 eqid 2460 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
875, 86zrh1 18310 . . . . . . . . . . 11  |-  ( R  e.  Ring  ->  ( ( ZRHom `  R ) `  1 )  =  ( 1r `  R
) )
8885, 87syl 16 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  1 )  =  ( 1r `  R ) )
8968, 83, 883eqtrd 2505 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } )  =  ( 1r `  R ) )
90 simp2l 1017 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  N  =  { I } )
9190mpteq1d 4521 . . . . . . . . . . 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 ) ) )
9291oveq2d 6291 . . . . . . . . . 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 ) ) ) )
938rngmgp 16985 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
9484, 93syl 16 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e.  Mnd )
95943ad2ant1 1012 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (mulGrp `  R )  e.  Mnd )
96 snidg 4046 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  I  e.  { I } )
9796adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  I  e.  { I } )
98 eleq2 2533 . . . . . . . . . . . . . . . . 17  |-  ( N  =  { I }  ->  ( I  e.  N  <->  I  e.  { I }
) )
9998adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( I  e.  N  <->  I  e.  { I } ) )
10097, 99mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  I  e.  N )
1013eleq2i 2538 . . . . . . . . . . . . . . . 16  |-  ( M  e.  B  <->  M  e.  ( Base `  A )
)
102101biimpi 194 . . . . . . . . . . . . . . 15  |-  ( M  e.  B  ->  M  e.  ( Base `  A
) )
103 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  N  /\  M  e.  ( Base `  A ) )  ->  I  e.  N )
104 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( I  e.  N  /\  M  e.  ( Base `  A ) )  ->  M  e.  ( Base `  A ) )
105103, 103, 1043jca 1171 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
106100, 102, 105syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( ( N  =  {
I }  /\  I  e.  V )  /\  M  e.  B )  ->  (
I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
1071063adant1 1009 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
108 eqid 2460 . . . . . . . . . . . . . 14  |-  ( Base `  R )  =  (
Base `  R )
1092, 108matecl 18687 . . . . . . . . . . . . 13  |-  ( ( I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( I M I )  e.  ( Base `  R ) )
110107, 109syl 16 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  R
) )
1118, 108mgpbas 16930 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
112110, 111syl6eleq 2558 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  (mulGrp `  R ) ) )
113 eqid 2460 . . . . . . . . . . . 12  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
114 fveq2 5857 . . . . . . . . . . . . . 14  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  x )  =  ( { <. I ,  I >. } `  I ) )
115 eqvisset 3114 . . . . . . . . . . . . . . 15  |-  ( x  =  I  ->  I  e.  _V )
116 fvsng 6086 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  _V  /\  I  e.  _V )  ->  ( { <. I ,  I >. } `  I
)  =  I )
117115, 115, 116syl2anc 661 . . . . . . . . . . . . . 14  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  I )  =  I )
118114, 117eqtrd 2501 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  x )  =  I )
119 id 22 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  x  =  I )
120118, 119oveq12d 6293 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
( { <. I ,  I >. } `  x
) M x )  =  ( I M I ) )
121113, 120gsumsn 16765 . . . . . . . . . . 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 ) )
12295, 15, 112, 121syl3anc 1223 . . . . . . . . . 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 ) )
12392, 122eqtrd 2501 . . . . . . . . 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 ) )
12489, 123oveq12d 6293 . . . . . . . 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 ) ) )
1251003ad2ant2 1013 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  I  e.  N )
1261023ad2ant3 1014 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  M  e.  ( Base `  A
) )
127125, 125, 126, 109syl3anc 1223 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  R
) )
128108, 7, 86rnglidm 17002 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
I M I )  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) ( I M I ) )  =  ( I M I ) )
12985, 127, 128syl2anc 661 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( 1r `  R
) ( .r `  R ) ( I M I ) )  =  ( I M I ) )
130124, 129eqtrd 2501 . . . . . . 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 ) )
131130opeq2d 4213 . . . . . 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 ) >. )
132131sneqd 4032 . . . . 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 ) >. } )
133 ovex 6300 . . . . . 6  |-  ( I M I )  e. 
_V
134 eqidd 2461 . . . . . . 7  |-  ( y  =  { <. I ,  I >. }  ->  (
I M I )  =  ( I M I ) )
135134fmptsng 6073 . . . . . 6  |-  ( ( { <. I ,  I >. }  e.  _V  /\  ( I M I )  e.  _V )  ->  { <. { <. I ,  I >. } ,  ( I M I )
>. }  =  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )
13624, 133, 135sylancl 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 ) ) )
137132, 136eqtrd 2501 . . . 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 ) ) )
13822, 34, 1373eqtrd 2505 . . 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 ) ) )
139138oveq2d 6291 . 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 ) ) ) )
140 rngmnd 16988 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
14184, 140syl 16 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Mnd )
1421413ad2ant1 1012 . . 3  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  R  e.  Mnd )
143108, 134gsumsn 16765 . . 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 ) )
144142, 24, 127, 143syl3anc 1223 . 2  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( R  gsumg  ( y  e.  { { <. I ,  I >. } }  |->  ( I M I ) ) )  =  ( I M I ) )
14510, 139, 1443eqtrd 2505 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 968    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106    \ cdif 3466   (/)c0 3778   ifcif 3932   {csn 4020   <.cop 4026    |-> cmpt 4498    _I cid 4783    X. cxp 4990   dom cdm 4992    |` cres 4994    o. ccom 4996    Fn wfn 5574   ` cfv 5579  (class class class)co 6275   Fincfn 7506   1c1 9482   Basecbs 14479   .rcmulr 14545    gsumg cgsu 14685   Mndcmnd 15715   SymGrpcsymg 16190  pmSgncpsgn 16303  mulGrpcmgp 16924   1rcur 16936   Ringcrg 16979   CRingccrg 16980   ZRHomczrh 18297   Mat cmat 18669   maDet cmdat 18846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-substr 12499  df-splice 12500  df-reverse 12501  df-s2 12763  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-0g 14686  df-gsum 14687  df-prds 14692  df-pws 14694  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-mulg 15854  df-subg 15986  df-ghm 16053  df-gim 16095  df-cntz 16143  df-oppg 16169  df-symg 16191  df-pmtr 16256  df-psgn 16305  df-cmn 16589  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-rnghom 17141  df-subrg 17203  df-sra 17594  df-rgmod 17595  df-cnfld 18185  df-zring 18250  df-zrh 18301  df-dsmm 18523  df-frlm 18538  df-mat 18670  df-mdet 18847
This theorem is referenced by:  chpmat1d  19097
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