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Theorem m2detleib 19315
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 2400 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2400 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2400 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 m2detleib.t . . . 4  |-  .x.  =  ( .r `  R )
8 eqid 2400 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib1 19275 . . 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 464 . 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 2400 . . 3  |-  ( Base `  R )  =  (
Base `  R )
12 eqid 2400 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
13 ringcmn 17439 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1413adantr 463 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e. CMnd )
15 m2detleib.n . . . . . 6  |-  N  =  { 1 ,  2 }
16 prfi 7747 . . . . . 6  |-  { 1 ,  2 }  e.  Fin
1715, 16eqeltri 2484 . . . . 5  |-  N  e. 
Fin
18 eqid 2400 . . . . . 6  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
1918, 4symgbasfi 16625 . . . . 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 455 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Ring )
2322adantr 463 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
244, 6, 5zrhpsgnelbas 18818 . . . . . 6  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2517, 24mp3an2 1312 . . . . 5  |-  ( ( R  e.  Ring  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2625adantlr 713 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
27 simpr 459 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  k  e.  (
Base `  ( SymGrp `  N ) ) )
28 simpr 459 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  B )
2928adantr 463 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  M  e.  B
)
3015, 4, 2, 3, 8m2detleiblem2 19312 . . . . 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 1228 . . . 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 17422 . . . 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 1228 . . 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 4652 . . . . . . . 8  |-  <. 1 ,  1 >.  e.  _V
35 opex 4652 . . . . . . . 8  |-  <. 2 ,  2 >.  e.  _V
3634, 35pm3.2i 453 . . . . . . 7  |-  ( <.
1 ,  1 >.  e.  _V  /\  <. 2 ,  2 >.  e.  _V )
37 opex 4652 . . . . . . . 8  |-  <. 1 ,  2 >.  e.  _V
38 opex 4652 . . . . . . . 8  |-  <. 2 ,  1 >.  e.  _V
3937, 38pm3.2i 453 . . . . . . 7  |-  ( <.
1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V )
4036, 39pm3.2i 453 . . . . . 6  |-  ( (
<. 1 ,  1
>.  e.  _V  /\  <. 2 ,  2 >.  e. 
_V )  /\  ( <. 1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V ) )
41 1ne2 10707 . . . . . . . . . 10  |-  1  =/=  2
4241olci 389 . . . . . . . . 9  |-  ( 1  =/=  1  \/  1  =/=  2 )
43 1ex 9539 . . . . . . . . . 10  |-  1  e.  _V
4443, 43opthne 4668 . . . . . . . . 9  |-  ( <.
1 ,  1 >.  =/=  <. 1 ,  2
>. 
<->  ( 1  =/=  1  \/  1  =/=  2
) )
4542, 44mpbir 209 . . . . . . . 8  |-  <. 1 ,  1 >.  =/=  <. 1 ,  2 >.
4641orci 388 . . . . . . . . 9  |-  ( 1  =/=  2  \/  1  =/=  1 )
4743, 43opthne 4668 . . . . . . . . 9  |-  ( <.
1 ,  1 >.  =/=  <. 2 ,  1
>. 
<->  ( 1  =/=  2  \/  1  =/=  1
) )
4846, 47mpbir 209 . . . . . . . 8  |-  <. 1 ,  1 >.  =/=  <. 2 ,  1 >.
4945, 48pm3.2i 453 . . . . . . 7  |-  ( <.
1 ,  1 >.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )
5049orci 388 . . . . . 6  |-  ( (
<. 1 ,  1
>.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) )
5140, 50pm3.2i 453 . . . . 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 4150 . . . . 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 427 . . . 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 4030 . . . 4  |-  ( {
<. 1 ,  1
>. ,  <. 2 ,  2 >. }  =/=  { <. 1 ,  2 >. ,  <. 2 ,  1
>. }  ->  ( { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  i^i  { { <. 1 ,  2
>. ,  <. 2 ,  1 >. } } )  =  (/) )
5652, 54, 553syl 20 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  ( { { <. 1 ,  1
>. ,  <. 2 ,  2 >. } }  i^i  { { <. 1 ,  2
>. ,  <. 2 ,  1 >. } } )  =  (/) )
57 2nn 10652 . . . . . 6  |-  2  e.  NN
5818, 4, 15symg2bas 16637 . . . . . 6  |-  ( ( 1  e.  _V  /\  2  e.  NN )  ->  ( Base `  ( SymGrp `
 N ) )  =  { { <. 1 ,  1 >. , 
<. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } } )
5943, 57, 58mp2an 670 . . . . 5  |-  ( Base `  ( SymGrp `  N )
)  =  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
60 df-pr 3972 . . . . 5  |-  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  =  ( { { <. 1 ,  1 >. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } } )
6159, 60eqtri 2429 . . . 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 17163 . 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 17417 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
6564adantr 463 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Mnd )
66 prex 4630 . . . . . 6  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V
6766a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V )
6866prid1 4077 . . . . . . . . 9  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
6968, 59eleqtrri 2487 . . . . . . . 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 18818 . . . . . . . 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 1312 . . . . . . 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 472 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  e.  ( Base `  R
) )
7415, 4, 2, 3, 8m2detleiblem2 19312 . . . . . . 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 1312 . . . . . 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 17422 . . . . . 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 1228 . . . . 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 5803 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )
7978fveq2d 5807 . . . . . . 7  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) )
80 fveq1 5802 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( k `  n )  =  ( { <. 1 ,  1
>. ,  <. 2 ,  2 >. } `  n
) )
8180oveq1d 6247 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) )
8281mpteq2dv 4479 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) )
8382oveq2d 6248 . . . . . . 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 6250 . . . . . 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 17192 . . . . 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 1228 . . . 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 4630 . . . . . 6  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V
8887a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V )
8987prid2 4078 . . . . . . . . 9  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
9089, 59eleqtrri 2487 . . . . . . . 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 18818 . . . . . . . 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 1312 . . . . . . 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 472 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  e.  ( Base `  R
) )
9515, 4, 2, 3, 8m2detleiblem2 19312 . . . . . . 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 1312 . . . . . 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 17422 . . . . . 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 1228 . . . . 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 5803 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )
10099fveq2d 5807 . . . . . . 7  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) )
101 fveq1 5802 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( k `  n )  =  ( { <. 1 ,  2
>. ,  <. 2 ,  1 >. } `  n
) )
102101oveq1d 6247 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) )
103102mpteq2dv 4479 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) )
104103oveq2d 6248 . . . . . . 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 6250 . . . . . 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 17192 . . . . 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 1228 . . . 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 6250 . . 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 2401 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
110 eqid 2400 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
11115, 4, 5, 6, 110m2detleiblem5 19309 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  1 >. ,  <. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  =  ( 1r `  R
) )
112109, 111sylan2 472 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  =  ( 1r `  R
) )
113 eqidd 2401 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
1148, 7mgpplusg 17355 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
11515, 4, 2, 3, 8, 114m2detleiblem3 19313 . . . . . . 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 1228 . . . . . 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 6250 . . . . 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 4077 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
119118, 15eleqtrri 2487 . . . . . . . . 9  |-  1  e.  N
120119a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  1  e.  N )
1213eleq2i 2478 . . . . . . . . . 10  |-  ( M  e.  B  <->  M  e.  ( Base `  A )
)
122121biimpi 194 . . . . . . . . 9  |-  ( M  e.  B  ->  M  e.  ( Base `  A
) )
123122adantl 464 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  ( Base `  A
) )
1242, 11matecl 19109 . . . . . . . 8  |-  ( ( 1  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 1 )  e.  ( Base `  R ) )
125120, 120, 123, 124syl3anc 1228 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 1 )  e.  ( Base `  R
) )
126 prid2g 4076 . . . . . . . . . . 11  |-  ( 2  e.  NN  ->  2  e.  { 1 ,  2 } )
12757, 126ax-mp 5 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
128127, 15eleqtrri 2487 . . . . . . . . 9  |-  2  e.  N
129128a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  2  e.  N )
1302, 11matecl 19109 . . . . . . . 8  |-  ( ( 2  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 2 )  e.  ( Base `  R ) )
131129, 129, 123, 130syl3anc 1228 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 2 )  e.  ( Base `  R
) )
13211, 7ringcl 17422 . . . . . . 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 1228 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )
13411, 7, 110ringlidm 17432 . . . . . 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 468 . . . . 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 2441 . . . 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 2401 . . . . . 6  |-  ( M  e.  B  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
138 eqid 2400 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
13915, 4, 5, 6, 110, 138m2detleiblem6 19310 . . . . . 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 472 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
141 eqidd 2401 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
14215, 4, 2, 3, 8, 114m2detleiblem4 19314 . . . . . 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 1228 . . . . 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 6250 . . . 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 6250 . . 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 19109 . . . . . 6  |-  ( ( 2  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 1 )  e.  ( Base `  R ) )
147129, 120, 123, 146syl3anc 1228 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 1 )  e.  ( Base `  R
) )
1482, 11matecl 19109 . . . . . 6  |-  ( ( 1  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 2 )  e.  ( Base `  R ) )
149120, 129, 123, 148syl3anc 1228 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 2 )  e.  ( Base `  R
) )
15011, 7ringcl 17422 . . . . 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 1228 . . . 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 19311 . . . 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 1228 . . 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 2445 . 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 2445 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 366    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   _Vcvv 3056    u. cun 3409    i^i cin 3410   (/)c0 3735   {csn 3969   {cpr 3971   <.cop 3975    |-> cmpt 4450   ` cfv 5523  (class class class)co 6232   Fincfn 7472   1c1 9441   NNcn 10494   2c2 10544   Basecbs 14731   +g cplusg 14799   .rcmulr 14800    gsumg cgsu 14945   Mndcmnd 16133   invgcminusg 16268   -gcsg 16269   SymGrpcsymg 16616  pmSgncpsgn 16728  CMndccmn 17012  mulGrpcmgp 17351   1rcur 17363   Ringcrg 17408   ZRHomczrh 18727   Mat cmat 19091   maDet cmdat 19268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-xor 1365  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-ot 3978  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-tpos 6910  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-rp 11182  df-fz 11642  df-fzo 11766  df-seq 12060  df-exp 12119  df-fac 12306  df-bc 12333  df-hash 12358  df-word 12496  df-lsw 12497  df-concat 12498  df-s1 12499  df-substr 12500  df-splice 12501  df-reverse 12502  df-s2 12774  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-0g 14946  df-gsum 14947  df-prds 14952  df-pws 14954  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-mhm 16180  df-submnd 16181  df-grp 16271  df-minusg 16272  df-sbg 16273  df-mulg 16274  df-subg 16412  df-ghm 16479  df-gim 16521  df-cntz 16569  df-oppg 16595  df-symg 16617  df-pmtr 16681  df-psgn 16730  df-cmn 17014  df-abl 17015  df-mgp 17352  df-ur 17364  df-ring 17410  df-cring 17411  df-rnghom 17574  df-subrg 17637  df-sra 18028  df-rgmod 18029  df-cnfld 18631  df-zring 18699  df-zrh 18731  df-dsmm 18951  df-frlm 18966  df-mat 19092  df-mdet 19269
This theorem is referenced by: (None)
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