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Theorem m2detleib 19733
Description: Leibniz' Formula for 2x2-matrices. (Contributed by AV, 21-Dec-2018.) (Revised by AV, 26-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
Hypotheses
Ref Expression
m2detleib.n  |-  N  =  { 1 ,  2 }
m2detleib.d  |-  D  =  ( N maDet  R )
m2detleib.a  |-  A  =  ( N Mat  R )
m2detleib.b  |-  B  =  ( Base `  A
)
m2detleib.m  |-  .-  =  ( -g `  R )
m2detleib.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
m2detleib  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( D `  M )  =  ( ( ( 1 M 1 ) 
.x.  ( 2 M 2 ) )  .-  ( ( 2 M 1 )  .x.  (
1 M 2 ) ) ) )

Proof of Theorem m2detleib
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 m2detleib.d . . . 4  |-  D  =  ( N maDet  R )
2 m2detleib.a . . . 4  |-  A  =  ( N Mat  R )
3 m2detleib.b . . . 4  |-  B  =  ( Base `  A
)
4 eqid 2471 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2471 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2471 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 m2detleib.t . . . 4  |-  .x.  =  ( .r `  R )
8 eqid 2471 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib1 19693 . . 3  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( k  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) )
109adantl 473 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( D `  M )  =  ( R  gsumg  ( k  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) )
11 eqid 2471 . . 3  |-  ( Base `  R )  =  (
Base `  R )
12 eqid 2471 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
13 ringcmn 17889 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1413adantr 472 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e. CMnd )
15 m2detleib.n . . . . . 6  |-  N  =  { 1 ,  2 }
16 prfi 7864 . . . . . 6  |-  { 1 ,  2 }  e.  Fin
1715, 16eqeltri 2545 . . . . 5  |-  N  e. 
Fin
18 eqid 2471 . . . . . 6  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
1918, 4symgbasfi 17105 . . . . 5  |-  ( N  e.  Fin  ->  ( Base `  ( SymGrp `  N
) )  e.  Fin )
2017, 19ax-mp 5 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  e.  Fin
2120a1i 11 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  e.  Fin )
22 simpl 464 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Ring )
2322adantr 472 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
244, 6, 5zrhpsgnelbas 19239 . . . . . 6  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2517, 24mp3an2 1378 . . . . 5  |-  ( ( R  e.  Ring  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2625adantlr 729 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
27 simpr 468 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  k  e.  (
Base `  ( SymGrp `  N ) ) )
28 simpr 468 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  B )
2928adantr 472 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  M  e.  B
)
3015, 4, 2, 3, 8m2detleiblem2 19730 . . . . 5  |-  ( ( R  e.  Ring  /\  k  e.  ( Base `  ( SymGrp `
 N ) )  /\  M  e.  B
)  ->  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) )  e.  ( Base `  R
) )
3123, 27, 29, 30syl3anc 1292 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) )  e.  ( Base `  R
) )
3211, 7ringcl 17872 . . . 4  |-  ( ( R  e.  Ring  /\  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  k
) )  e.  (
Base `  R )  /\  ( (mulGrp `  R
)  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) )  e.  ( Base `  R
) )  ->  (
( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) )  e.  ( Base `  R
) )
3323, 26, 31, 32syl3anc 1292 . . 3  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  ( ( ( ZRHom `  R ) `  ( (pmSgn `  N
) `  k )
)  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) ) )  e.  ( Base `  R
) )
34 opex 4664 . . . . . . . 8  |-  <. 1 ,  1 >.  e.  _V
35 opex 4664 . . . . . . . 8  |-  <. 2 ,  2 >.  e.  _V
3634, 35pm3.2i 462 . . . . . . 7  |-  ( <.
1 ,  1 >.  e.  _V  /\  <. 2 ,  2 >.  e.  _V )
37 opex 4664 . . . . . . . 8  |-  <. 1 ,  2 >.  e.  _V
38 opex 4664 . . . . . . . 8  |-  <. 2 ,  1 >.  e.  _V
3937, 38pm3.2i 462 . . . . . . 7  |-  ( <.
1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V )
4036, 39pm3.2i 462 . . . . . 6  |-  ( (
<. 1 ,  1
>.  e.  _V  /\  <. 2 ,  2 >.  e. 
_V )  /\  ( <. 1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V ) )
41 1ne2 10845 . . . . . . . . . 10  |-  1  =/=  2
4241olci 398 . . . . . . . . 9  |-  ( 1  =/=  1  \/  1  =/=  2 )
43 1ex 9656 . . . . . . . . . 10  |-  1  e.  _V
4443, 43opthne 4682 . . . . . . . . 9  |-  ( <.
1 ,  1 >.  =/=  <. 1 ,  2
>. 
<->  ( 1  =/=  1  \/  1  =/=  2
) )
4542, 44mpbir 214 . . . . . . . 8  |-  <. 1 ,  1 >.  =/=  <. 1 ,  2 >.
4641orci 397 . . . . . . . . 9  |-  ( 1  =/=  2  \/  1  =/=  1 )
4743, 43opthne 4682 . . . . . . . . 9  |-  ( <.
1 ,  1 >.  =/=  <. 2 ,  1
>. 
<->  ( 1  =/=  2  \/  1  =/=  1
) )
4846, 47mpbir 214 . . . . . . . 8  |-  <. 1 ,  1 >.  =/=  <. 2 ,  1 >.
4945, 48pm3.2i 462 . . . . . . 7  |-  ( <.
1 ,  1 >.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )
5049orci 397 . . . . . 6  |-  ( (
<. 1 ,  1
>.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) )
5140, 50pm3.2i 462 . . . . 5  |-  ( ( ( <. 1 ,  1
>.  e.  _V  /\  <. 2 ,  2 >.  e. 
_V )  /\  ( <. 1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V ) )  /\  (
( <. 1 ,  1
>.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) ) )
5251a1i 11 . . . 4  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( <. 1 ,  1 >.  e.  _V  /\ 
<. 2 ,  2
>.  e.  _V )  /\  ( <. 1 ,  2
>.  e.  _V  /\  <. 2 ,  1 >.  e. 
_V ) )  /\  ( ( <. 1 ,  1 >.  =/=  <. 1 ,  2 >.  /\ 
<. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) ) ) )
53 prneimg 4148 . . . . 5  |-  ( ( ( <. 1 ,  1
>.  e.  _V  /\  <. 2 ,  2 >.  e. 
_V )  /\  ( <. 1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V ) )  ->  (
( ( <. 1 ,  1 >.  =/=  <. 1 ,  2 >.  /\ 
<. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =/=  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } ) )
5453imp 436 . . . 4  |-  ( ( ( ( <. 1 ,  1 >.  e.  _V  /\ 
<. 2 ,  2
>.  e.  _V )  /\  ( <. 1 ,  2
>.  e.  _V  /\  <. 2 ,  1 >.  e. 
_V ) )  /\  ( ( <. 1 ,  1 >.  =/=  <. 1 ,  2 >.  /\ 
<. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) ) )  ->  { <. 1 ,  1
>. ,  <. 2 ,  2 >. }  =/=  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } )
55 disjsn2 4024 . . . 4  |-  ( {
<. 1 ,  1
>. ,  <. 2 ,  2 >. }  =/=  { <. 1 ,  2 >. ,  <. 2 ,  1
>. }  ->  ( { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  i^i  { { <. 1 ,  2
>. ,  <. 2 ,  1 >. } } )  =  (/) )
5652, 54, 553syl 18 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  i^i  { { <. 1 ,  2
>. ,  <. 2 ,  1 >. } } )  =  (/) )
57 2nn 10790 . . . . . 6  |-  2  e.  NN
5818, 4, 15symg2bas 17117 . . . . . 6  |-  ( ( 1  e.  _V  /\  2  e.  NN )  ->  ( Base `  ( SymGrp `
 N ) )  =  { { <. 1 ,  1 >. , 
<. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } } )
5943, 57, 58mp2an 686 . . . . 5  |-  ( Base `  ( SymGrp `  N )
)  =  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
60 df-pr 3962 . . . . 5  |-  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  =  ( { { <. 1 ,  1 >. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } } )
6159, 60eqtri 2493 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2
>. ,  <. 2 ,  1 >. } } )
6261a1i 11 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( Base `  ( SymGrp `  N
) )  =  ( { { <. 1 ,  1 >. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } } ) )
6311, 12, 14, 21, 33, 56, 62gsummptfidmsplit 17641 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( R  gsumg  ( k  e.  (
Base `  ( SymGrp `  N ) )  |->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) )  =  ( ( R  gsumg  ( k  e.  { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  |->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) ( +g  `  R
) ( R  gsumg  ( k  e.  { { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  |->  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  k
) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) ) )
64 ringmnd 17867 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
6564adantr 472 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Mnd )
66 prex 4642 . . . . . 6  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V
6766a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V )
6866prid1 4071 . . . . . . . . 9  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
6968, 59eleqtrri 2548 . . . . . . . 8  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  ( Base `  ( SymGrp `  N )
)
7069a1i 11 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  ( Base `  ( SymGrp `  N )
) )
714, 6, 5zrhpsgnelbas 19239 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  ( Base `  ( SymGrp `  N )
) )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  e.  ( Base `  R
) )
7217, 71mp3an2 1378 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  1 >. ,  <. 2 ,  2
>. }  e.  ( Base `  ( SymGrp `  N )
) )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  e.  ( Base `  R
) )
7370, 72sylan2 482 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  e.  ( Base `  R
) )
7415, 4, 2, 3, 8m2detleiblem2 19730 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  1 >. ,  <. 2 ,  2
>. }  e.  ( Base `  ( SymGrp `  N )
)  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) )  e.  ( Base `  R
) )
7569, 74mp3an2 1378 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) )  e.  ( Base `  R
) )
7611, 7ringcl 17872 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  e.  ( Base `  R
)  /\  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) )  e.  ( Base `  R ) )  -> 
( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) )  e.  (
Base `  R )
)
7722, 73, 75, 76syl3anc 1292 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) )  e.  (
Base `  R )
)
78 fveq2 5879 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )
7978fveq2d 5883 . . . . . . 7  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) )
80 fveq1 5878 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( k `  n )  =  ( { <. 1 ,  1
>. ,  <. 2 ,  2 >. } `  n
) )
8180oveq1d 6323 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) )
8281mpteq2dv 4483 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) )
8382oveq2d 6324 . . . . . . 7  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) )  =  ( (mulGrp `  R
)  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) )
8479, 83oveq12d 6326 . . . . . 6  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( ( ZRHom `  R ) `  ( (pmSgn `  N
) `  k )
)  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) ) )  =  ( ( ( ZRHom `  R ) `  ( (pmSgn `  N
) `  { <. 1 ,  1 >. ,  <. 2 ,  2 >. } ) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) ) ) )
8511, 84gsumsn 17665 . . . . 5  |-  ( ( R  e.  Mnd  /\  {
<. 1 ,  1
>. ,  <. 2 ,  2 >. }  e.  _V  /\  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) )  e.  (
Base `  R )
)  ->  ( R  gsumg  ( k  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } }  |->  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  k
) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) )  =  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) ) ) )
8665, 67, 77, 85syl3anc 1292 . . . 4  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( R  gsumg  ( k  e.  { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  |->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) )  =  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) ) ) )
87 prex 4642 . . . . . 6  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V
8887a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V )
8987prid2 4072 . . . . . . . . 9  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
9089, 59eleqtrri 2548 . . . . . . . 8  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  ( Base `  ( SymGrp `  N )
)
9190a1i 11 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  ( Base `  ( SymGrp `  N )
) )
924, 6, 5zrhpsgnelbas 19239 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  ( Base `  ( SymGrp `  N )
) )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  e.  ( Base `  R
) )
9317, 92mp3an2 1378 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  2 >. ,  <. 2 ,  1
>. }  e.  ( Base `  ( SymGrp `  N )
) )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  e.  ( Base `  R
) )
9491, 93sylan2 482 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  e.  ( Base `  R
) )
9515, 4, 2, 3, 8m2detleiblem2 19730 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  2 >. ,  <. 2 ,  1
>. }  e.  ( Base `  ( SymGrp `  N )
)  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) )  e.  ( Base `  R
) )
9690, 95mp3an2 1378 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) )  e.  ( Base `  R
) )
9711, 7ringcl 17872 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  e.  ( Base `  R
)  /\  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) )  e.  ( Base `  R ) )  -> 
( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) ) )  e.  (
Base `  R )
)
9822, 94, 96, 97syl3anc 1292 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) ) )  e.  (
Base `  R )
)
99 fveq2 5879 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )
10099fveq2d 5883 . . . . . . 7  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) )
101 fveq1 5878 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( k `  n )  =  ( { <. 1 ,  2
>. ,  <. 2 ,  1 >. } `  n
) )
102101oveq1d 6323 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) )
103102mpteq2dv 4483 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) )
104103oveq2d 6324 . . . . . . 7  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) )  =  ( (mulGrp `  R
)  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) ) )
105100, 104oveq12d 6326 . . . . . 6  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( ( ZRHom `  R ) `  ( (pmSgn `  N
) `  k )
)  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n ) M n ) ) ) )  =  ( ( ( ZRHom `  R ) `  ( (pmSgn `  N
) `  { <. 1 ,  2 >. ,  <. 2 ,  1 >. } ) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) ) ) )
10611, 105gsumsn 17665 . . . . 5  |-  ( ( R  e.  Mnd  /\  {
<. 1 ,  2
>. ,  <. 2 ,  1 >. }  e.  _V  /\  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) ) )  e.  (
Base `  R )
)  ->  ( R  gsumg  ( k  e.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } }  |->  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  k
) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) )  =  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) ) ) )
10765, 88, 98, 106syl3anc 1292 . . . 4  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( R  gsumg  ( k  e.  { { <. 1 ,  2
>. ,  <. 2 ,  1 >. } }  |->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) )  =  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) ) ) )
10886, 107oveq12d 6326 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( R  gsumg  ( k  e.  { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  |->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) ( +g  `  R
) ( R  gsumg  ( k  e.  { { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  |->  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  k
) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) )  =  ( ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) ) ( +g  `  R ) ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) ) ) ) )
109 eqidd 2472 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
110 eqid 2471 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
11115, 4, 5, 6, 110m2detleiblem5 19727 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  1 >. ,  <. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  =  ( 1r `  R
) )
112109, 111sylan2 482 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  =  ( 1r `  R
) )
113 eqidd 2472 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
1148, 7mgpplusg 17805 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
11515, 4, 2, 3, 8, 114m2detleiblem3 19731 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  1 >. ,  <. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) )  =  ( ( 1 M 1 )  .x.  ( 2 M 2 ) ) )
11622, 113, 28, 115syl3anc 1292 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) )  =  ( ( 1 M 1 )  .x.  ( 2 M 2 ) ) )
117112, 116oveq12d 6326 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) )  =  ( ( 1r `  R
)  .x.  ( (
1 M 1 ) 
.x.  ( 2 M 2 ) ) ) )
11843prid1 4071 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
119118, 15eleqtrri 2548 . . . . . . . . 9  |-  1  e.  N
120119a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  1  e.  N )
1213eleq2i 2541 . . . . . . . . . 10  |-  ( M  e.  B  <->  M  e.  ( Base `  A )
)
122121biimpi 199 . . . . . . . . 9  |-  ( M  e.  B  ->  M  e.  ( Base `  A
) )
123122adantl 473 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  ( Base `  A
) )
1242, 11matecl 19527 . . . . . . . 8  |-  ( ( 1  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 1 )  e.  ( Base `  R ) )
125120, 120, 123, 124syl3anc 1292 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 1 )  e.  ( Base `  R
) )
126 prid2g 4070 . . . . . . . . . . 11  |-  ( 2  e.  NN  ->  2  e.  { 1 ,  2 } )
12757, 126ax-mp 5 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
128127, 15eleqtrri 2548 . . . . . . . . 9  |-  2  e.  N
129128a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  2  e.  N )
1302, 11matecl 19527 . . . . . . . 8  |-  ( ( 2  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 2 )  e.  ( Base `  R ) )
131129, 129, 123, 130syl3anc 1292 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 2 )  e.  ( Base `  R
) )
13211, 7ringcl 17872 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
1 M 1 )  e.  ( Base `  R
)  /\  ( 2 M 2 )  e.  ( Base `  R
) )  ->  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )
13322, 125, 131, 132syl3anc 1292 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )
13411, 7, 110ringlidm 17882 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )  ->  (
( 1r `  R
)  .x.  ( (
1 M 1 ) 
.x.  ( 2 M 2 ) ) )  =  ( ( 1 M 1 )  .x.  ( 2 M 2 ) ) )
135133, 134syldan 478 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 1r `  R
)  .x.  ( (
1 M 1 ) 
.x.  ( 2 M 2 ) ) )  =  ( ( 1 M 1 )  .x.  ( 2 M 2 ) ) )
136117, 135eqtrd 2505 . . . 4  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) )  =  ( ( 1 M 1 )  .x.  ( 2 M 2 ) ) )
137 eqidd 2472 . . . . . 6  |-  ( M  e.  B  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
138 eqid 2471 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
13915, 4, 5, 6, 110, 138m2detleiblem6 19728 . . . . . 6  |-  ( ( R  e.  Ring  /\  { <. 1 ,  2 >. ,  <. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
140137, 139sylan2 482 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
141 eqidd 2472 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
14215, 4, 2, 3, 8, 114m2detleiblem4 19732 . . . . . 6  |-  ( ( R  e.  Ring  /\  { <. 1 ,  2 >. ,  <. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) )  =  ( ( 2 M 1 )  .x.  ( 1 M 2 ) ) )
14322, 141, 28, 142syl3anc 1292 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) )  =  ( ( 2 M 1 )  .x.  ( 1 M 2 ) ) )
144140, 143oveq12d 6326 . . . 4  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) ) )  =  ( ( ( invg `  R ) `  ( 1r `  R ) ) 
.x.  ( ( 2 M 1 )  .x.  ( 1 M 2 ) ) ) )
145136, 144oveq12d 6326 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) 
.x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) ) ) ( +g  `  R ) ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) ) ) )  =  ( ( ( 1 M 1 )  .x.  (
2 M 2 ) ) ( +g  `  R
) ( ( ( invg `  R
) `  ( 1r `  R ) )  .x.  ( ( 2 M 1 )  .x.  (
1 M 2 ) ) ) ) )
1462, 11matecl 19527 . . . . . 6  |-  ( ( 2  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 1 )  e.  ( Base `  R ) )
147129, 120, 123, 146syl3anc 1292 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 1 )  e.  ( Base `  R
) )
1482, 11matecl 19527 . . . . . 6  |-  ( ( 1  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 2 )  e.  ( Base `  R ) )
149120, 129, 123, 148syl3anc 1292 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 2 )  e.  ( Base `  R
) )
15011, 7ringcl 17872 . . . . 5  |-  ( ( R  e.  Ring  /\  (
2 M 1 )  e.  ( Base `  R
)  /\  ( 1 M 2 )  e.  ( Base `  R
) )  ->  (
( 2 M 1 )  .x.  ( 1 M 2 ) )  e.  ( Base `  R
) )
15122, 147, 149, 150syl3anc 1292 . . . 4  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 2 M 1 )  .x.  ( 1 M 2 ) )  e.  ( Base `  R
) )
152 m2detleib.m . . . . 5  |-  .-  =  ( -g `  R )
15315, 4, 5, 6, 110, 138, 7, 152m2detleiblem7 19729 . . . 4  |-  ( ( R  e.  Ring  /\  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
)  /\  ( (
2 M 1 ) 
.x.  ( 1 M 2 ) )  e.  ( Base `  R
) )  ->  (
( ( 1 M 1 )  .x.  (
2 M 2 ) ) ( +g  `  R
) ( ( ( invg `  R
) `  ( 1r `  R ) )  .x.  ( ( 2 M 1 )  .x.  (
1 M 2 ) ) ) )  =  ( ( ( 1 M 1 )  .x.  ( 2 M 2 ) )  .-  (
( 2 M 1 )  .x.  ( 1 M 2 ) ) ) )
15422, 133, 151, 153syl3anc 1292 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ( 1 M 1 )  .x.  (
2 M 2 ) ) ( +g  `  R
) ( ( ( invg `  R
) `  ( 1r `  R ) )  .x.  ( ( 2 M 1 )  .x.  (
1 M 2 ) ) ) )  =  ( ( ( 1 M 1 )  .x.  ( 2 M 2 ) )  .-  (
( 2 M 1 )  .x.  ( 1 M 2 ) ) ) )
155108, 145, 1543eqtrd 2509 . 2  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( R  gsumg  ( k  e.  { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  |->  ( ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  .x.  ( (mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) ( +g  `  R
) ( R  gsumg  ( k  e.  { { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  |->  ( ( ( ZRHom `  R
) `  ( (pmSgn `  N ) `  k
) )  .x.  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( k `  n
) M n ) ) ) ) ) ) )  =  ( ( ( 1 M 1 )  .x.  (
2 M 2 ) )  .-  ( ( 2 M 1 ) 
.x.  ( 1 M 2 ) ) ) )
15610, 63, 1553eqtrd 2509 1  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( D `  M )  =  ( ( ( 1 M 1 ) 
.x.  ( 2 M 2 ) )  .-  ( ( 2 M 1 )  .x.  (
1 M 2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   {cpr 3961   <.cop 3965    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   Fincfn 7587   1c1 9558   NNcn 10631   2c2 10681   Basecbs 15199   +g cplusg 15268   .rcmulr 15269    gsumg cgsu 15417   Mndcmnd 16613   invgcminusg 16748   -gcsg 16749   SymGrpcsymg 17096  pmSgncpsgn 17208  CMndccmn 17508  mulGrpcmgp 17801   1rcur 17813   Ringcrg 17858   ZRHomczrh 19148   Mat cmat 19509   maDet cmdat 19686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-xor 1431  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-reverse 12717  df-s2 13003  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-gim 17001  df-cntz 17049  df-oppg 17075  df-symg 17097  df-pmtr 17161  df-psgn 17210  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-rnghom 18021  df-subrg 18084  df-sra 18473  df-rgmod 18474  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-dsmm 19372  df-frlm 19387  df-mat 19510  df-mdet 19687
This theorem is referenced by:  lmat22det  28722
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