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Theorem m1detdiag 18528
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 2451 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2451 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2451 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 eqid 2451 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2451 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib 18518 . . 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 1011 . 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 5792 . . . . . . . . 9  |-  ( N  =  { I }  ->  ( SymGrp `  N )  =  ( SymGrp `  {
I } ) )
1211fveq2d 5796 . . . . . . . 8  |-  ( N  =  { I }  ->  ( Base `  ( SymGrp `
 N ) )  =  ( Base `  ( SymGrp `
 { I }
) ) )
1312adantr 465 . . . . . . 7  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  { I } ) ) )
14133ad2ant2 1010 . . . . . 6  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  (
Base `  ( SymGrp `  { I } ) ) )
15 simp2r 1015 . . . . . . 7  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  I  e.  V )
16 eqid 2451 . . . . . . . 8  |-  ( SymGrp `  { I } )  =  ( SymGrp `  {
I } )
17 eqid 2451 . . . . . . . 8  |-  ( Base `  ( SymGrp `  { I } ) )  =  ( Base `  ( SymGrp `
 { I }
) )
18 eqid 2451 . . . . . . . 8  |-  { I }  =  { I }
1916, 17, 18symg1bas 16012 . . . . . . 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 2492 . . . . 5  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  { { <. I ,  I >. } } )
2221mpteq1d 4474 . . . 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 4634 . . . . . 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 6218 . . . . 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 5792 . . . . . . . 8  |-  ( p  =  { <. I ,  I >. }  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } ) )
27 fveq1 5791 . . . . . . . . . . 11  |-  ( p  =  { <. I ,  I >. }  ->  (
p `  x )  =  ( { <. I ,  I >. } `  x ) )
2827oveq1d 6208 . . . . . . . . . 10  |-  ( p  =  { <. I ,  I >. }  ->  (
( p `  x
) M x )  =  ( ( {
<. I ,  I >. } `
 x ) M x ) )
2928mpteq2dv 4480 . . . . . . . . 9  |-  ( p  =  { <. I ,  I >. }  ->  (
x  e.  N  |->  ( ( p `  x
) M x ) )  =  ( x  e.  N  |->  ( ( { <. I ,  I >. } `  x ) M x ) ) )
3029oveq2d 6209 . . . . . . . 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 6211 . . . . . . 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 6002 . . . . . 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 2459 . . . . 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 2451 . . . . . . . . . . . . 13  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
36 eqid 2451 . . . . . . . . . . . . 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 16118 . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  ( Base `  ( SymGrp `  N )
)  =  { { <. I ,  I >. } } )
40393ad2ant2 1010 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  { { <. I ,  I >. } } )
41 rabeq 3065 . . . . . . . . . . . . . . 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 3568 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  { <. I ,  I >. }  ->  (
b  \  _I  )  =  ( { <. I ,  I >. }  \  _I  ) )
4443dmeqd 5143 . . . . . . . . . . . . . . . . 17  |-  ( b  =  { <. I ,  I >. }  ->  dom  ( b  \  _I  )  =  dom  ( {
<. I ,  I >. } 
\  _I  ) )
4544eleq1d 2520 . . . . . . . . . . . . . . . 16  |-  ( b  =  { <. I ,  I >. }  ->  ( dom  ( b  \  _I  )  e.  Fin  <->  dom  ( {
<. I ,  I >. } 
\  _I  )  e. 
Fin ) )
4645rabsnif 4045 . . . . . . . . . . . . . . 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 5263 . . . . . . . . . . . . . . . . . . . 20  |-  (  _I  |`  { I } )  =  ( { I }  X.  { I }
)
49 xpsng 5986 . . . . . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( I  e.  V  ->  { <. I ,  I >. }  =  (  _I  |`  { I } ) )
52 fnsng 5566 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  e.  V  /\  I  e.  V )  ->  { <. I ,  I >. }  Fn  { I } )
5352anidms 645 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  V  ->  { <. I ,  I >. }  Fn  { I } )
54 fnnfpeq0 6011 . . . . . . . . . . . . . . . . . . . 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 7644 . . . . . . . . . . . . . . . . . 18  |-  (/)  e.  Fin
5856, 57syl6eqel 2547 . . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  dom  ( { <. I ,  I >. }  \  _I  )  e.  Fin )
61 iftrue 3898 . . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . 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 5603 . . . . . . . . . . . 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 4006 . . . . . . . . . . 11  |-  { <. I ,  I >. }  e.  { { <. I ,  I >. } }
67 fvco2 5868 . . . . . . . . . . 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 5792 . . . . . . . . . . . . . . 15  |-  ( N  =  { I }  ->  (pmSgn `  N )  =  (pmSgn `  { I } ) )
7069adantr 465 . . . . . . . . . . . . . 14  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  (pmSgn `  N
)  =  (pmSgn `  { I } ) )
71703ad2ant2 1010 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (pmSgn `  N )  =  (pmSgn `  { I } ) )
7271fveq1d 5794 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N ) `  { <. I ,  I >. } )  =  ( (pmSgn `  { I } ) `  { <. I ,  I >. } ) )
73 snidg 4004 . . . . . . . . . . . . . . . . . 18  |-  ( {
<. I ,  I >. }  e.  _V  ->  { <. I ,  I >. }  e.  { { <. I ,  I >. } } )
7423, 73mp1i 12 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  { <. I ,  I >. }  e.  { { <. I ,  I >. } } )
7574, 19eleqtrrd 2542 . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I  e.  V  /\  {
<. I ,  I >. }  e.  ( Base `  ( SymGrp `
 { I }
) ) ) )
79 eqid 2451 . . . . . . . . . . . . . 14  |-  (pmSgn `  { I } )  =  (pmSgn `  {
I } )
8018, 16, 17, 79psgnsn 16137 . . . . . . . . . . . . 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 2492 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
(pmSgn `  N ) `  { <. I ,  I >. } )  =  1 )
8382fveq2d 5796 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. I ,  I >. } ) )  =  ( ( ZRHom `  R
) `  1 )
)
84 crngrng 16770 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  R  e.  Ring )
85843ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  R  e.  Ring )
86 eqid 2451 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
875, 86zrh1 18062 . . . . . . . . . . 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 2496 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  { <. I ,  I >. } )  =  ( 1r `  R ) )
90 simp2l 1014 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  N  =  { I } )
9190mpteq1d 4474 . . . . . . . . . . 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 6209 . . . . . . . . . 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 16766 . . . . . . . . . . . . 13  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
9484, 93syl 16 . . . . . . . . . . . 12  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e.  Mnd )
95943ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (mulGrp `  R )  e.  Mnd )
96 snidg 4004 . . . . . . . . . . . . . . . . 17  |-  ( I  e.  V  ->  I  e.  { I } )
9796adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( N  =  { I }  /\  I  e.  V
)  ->  I  e.  { I } )
98 eleq2 2524 . . . . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . 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 1006 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I  e.  N  /\  I  e.  N  /\  M  e.  ( Base `  A ) ) )
108 eqid 2451 . . . . . . . . . . . . . 14  |-  ( Base `  R )  =  (
Base `  R )
1092, 108matecl 18444 . . . . . . . . . . . . 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 16711 . . . . . . . . . . . 12  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
112110, 111syl6eleq 2549 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  (mulGrp `  R ) ) )
113 eqid 2451 . . . . . . . . . . . 12  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
114 fveq2 5792 . . . . . . . . . . . . . 14  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  x )  =  ( { <. I ,  I >. } `  I ) )
115 eqvisset 3079 . . . . . . . . . . . . . . 15  |-  ( x  =  I  ->  I  e.  _V )
116 fvsng 6014 . . . . . . . . . . . . . . 15  |-  ( ( I  e.  _V  /\  I  e.  _V )  ->  ( { <. I ,  I >. } `  I
)  =  I )
117115, 115, 116syl2anc 661 . . . . . . . . . . . . . 14  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  I )  =  I )
118114, 117eqtrd 2492 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  ( { <. I ,  I >. } `  x )  =  I )
119 id 22 . . . . . . . . . . . . 13  |-  ( x  =  I  ->  x  =  I )
120118, 119oveq12d 6211 . . . . . . . . . . . 12  |-  ( x  =  I  ->  (
( { <. I ,  I >. } `  x
) M x )  =  ( I M I ) )
121113, 120gsumsn 16563 . . . . . . . . . . 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 1219 . . . . . . . . . 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 2492 . . . . . . . . 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 6211 . . . . . . . 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 1010 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  I  e.  N )
1261023ad2ant3 1011 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  M  e.  ( Base `  A
) )
127125, 125, 126, 109syl3anc 1219 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  (
I M I )  e.  ( Base `  R
) )
128108, 7, 86rnglidm 16783 . . . . . . . . 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 2492 . . . . . . 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 4167 . . . . . 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 3990 . . . . 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 6218 . . . . . 6  |-  ( I M I )  e. 
_V
134 eqidd 2452 . . . . . . 7  |-  ( y  =  { <. I ,  I >. }  ->  (
I M I )  =  ( I M I ) )
135134fmptsng 6002 . . . . . 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 2492 . . . 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 2496 . . 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 6209 . 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 16769 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
14184, 140syl 16 . . . 4  |-  ( R  e.  CRing  ->  R  e.  Mnd )
1421413ad2ant1 1009 . . 3  |-  ( ( R  e.  CRing  /\  ( N  =  { I }  /\  I  e.  V
)  /\  M  e.  B )  ->  R  e.  Mnd )
143108, 134gsumsn 16563 . . 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 1219 . 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 2496 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 965    = wceq 1370    e. wcel 1758   {crab 2799   _Vcvv 3071    \ cdif 3426   (/)c0 3738   ifcif 3892   {csn 3978   <.cop 3984    |-> cmpt 4451    _I cid 4732    X. cxp 4939   dom cdm 4941    |` cres 4943    o. ccom 4945    Fn wfn 5514   ` cfv 5519  (class class class)co 6193   Fincfn 7413   1c1 9387   Basecbs 14285   .rcmulr 14350    gsumg cgsu 14490   Mndcmnd 15520   SymGrpcsymg 15993  pmSgncpsgn 16106  mulGrpcmgp 16705   1rcur 16717   Ringcrg 16760   CRingccrg 16761   ZRHomczrh 18049   Mat cmat 18398   maDet cmdat 18515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-splice 12345  df-reverse 12346  df-s2 12586  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-0g 14491  df-gsum 14492  df-prds 14497  df-pws 14499  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-mulg 15659  df-subg 15789  df-ghm 15856  df-gim 15898  df-cntz 15946  df-oppg 15972  df-symg 15994  df-pmtr 16059  df-psgn 16108  df-cmn 16392  df-mgp 16706  df-ur 16718  df-rng 16762  df-cring 16763  df-rnghom 16921  df-subrg 16978  df-sra 17368  df-rgmod 17369  df-cnfld 17937  df-zring 18002  df-zrh 18053  df-dsmm 18275  df-frlm 18290  df-mat 18400  df-mdet 18516
This theorem is referenced by:  cpmat1d  31293
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