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Theorem m2detleib 18435
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 2441 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2441 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2441 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 m2detleib.t . . . 4  |-  .x.  =  ( .r `  R )
8 eqid 2441 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib1 18400 . . 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 466 . 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 2441 . . 3  |-  ( Base `  R )  =  (
Base `  R )
12 eqid 2441 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngcmn 16673 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1413adantr 465 . . 3  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e. CMnd )
15 m2detleib.n . . . . . 6  |-  N  =  { 1 ,  2 }
16 prfi 7584 . . . . . 6  |-  { 1 ,  2 }  e.  Fin
1715, 16eqeltri 2511 . . . . 5  |-  N  e. 
Fin
18 eqid 2441 . . . . . 6  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
1918, 4symgbasfi 15889 . . . . 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 457 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Ring )
2322adantr 465 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
244, 6, 5zrhpsgnelbas 18022 . . . . . 6  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2517, 24mp3an2 1302 . . . . 5  |-  ( ( R  e.  Ring  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2625adantlr 714 . . . 4  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
27 simpr 461 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  k  e.  (
Base `  ( SymGrp `  N ) ) )
28 simpr 461 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  B )
2928adantr 465 . . . . 5  |-  ( ( ( R  e.  Ring  /\  M  e.  B )  /\  k  e.  (
Base `  ( SymGrp `  N ) ) )  ->  M  e.  B
)
3015, 4, 2, 3, 8m2detleiblem2 18432 . . . . 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 1218 . . . 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, 7rngcl 16656 . . . 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 1218 . . 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 4554 . . . . . . . 8  |-  <. 1 ,  1 >.  e.  _V
35 opex 4554 . . . . . . . 8  |-  <. 2 ,  2 >.  e.  _V
3634, 35pm3.2i 455 . . . . . . 7  |-  ( <.
1 ,  1 >.  e.  _V  /\  <. 2 ,  2 >.  e.  _V )
37 opex 4554 . . . . . . . 8  |-  <. 1 ,  2 >.  e.  _V
38 opex 4554 . . . . . . . 8  |-  <. 2 ,  1 >.  e.  _V
3937, 38pm3.2i 455 . . . . . . 7  |-  ( <.
1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V )
4036, 39pm3.2i 455 . . . . . 6  |-  ( (
<. 1 ,  1
>.  e.  _V  /\  <. 2 ,  2 >.  e. 
_V )  /\  ( <. 1 ,  2 >.  e.  _V  /\  <. 2 ,  1 >.  e.  _V ) )
41 1ne2 10532 . . . . . . . . . 10  |-  1  =/=  2
4241olci 391 . . . . . . . . 9  |-  ( 1  =/=  1  \/  1  =/=  2 )
43 1ex 9379 . . . . . . . . . 10  |-  1  e.  _V
4443, 43opthne 4570 . . . . . . . . 9  |-  ( <.
1 ,  1 >.  =/=  <. 1 ,  2
>. 
<->  ( 1  =/=  1  \/  1  =/=  2
) )
4542, 44mpbir 209 . . . . . . . 8  |-  <. 1 ,  1 >.  =/=  <. 1 ,  2 >.
4641orci 390 . . . . . . . . 9  |-  ( 1  =/=  2  \/  1  =/=  1 )
4743, 43opthne 4570 . . . . . . . . 9  |-  ( <.
1 ,  1 >.  =/=  <. 2 ,  1
>. 
<->  ( 1  =/=  2  \/  1  =/=  1
) )
4846, 47mpbir 209 . . . . . . . 8  |-  <. 1 ,  1 >.  =/=  <. 2 ,  1 >.
4945, 48pm3.2i 455 . . . . . . 7  |-  ( <.
1 ,  1 >.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )
5049orci 390 . . . . . 6  |-  ( (
<. 1 ,  1
>.  =/=  <. 1 ,  2
>.  /\  <. 1 ,  1
>.  =/=  <. 2 ,  1
>. )  \/  ( <. 2 ,  2 >.  =/=  <. 1 ,  2
>.  /\  <. 2 ,  2
>.  =/=  <. 2 ,  1
>. ) )
5140, 50pm3.2i 455 . . . . 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 4051 . . . . 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 429 . . . 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 3935 . . . 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 10477 . . . . . 6  |-  2  e.  NN
5818, 4, 15symg2bas 15901 . . . . . 6  |-  ( ( 1  e.  _V  /\  2  e.  NN )  ->  ( Base `  ( SymGrp `
 N ) )  =  { { <. 1 ,  1 >. , 
<. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } } )
5943, 57, 58mp2an 672 . . . . 5  |-  ( Base `  ( SymGrp `  N )
)  =  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
60 df-pr 3878 . . . . 5  |-  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  =  ( { { <. 1 ,  1 >. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } } )
6159, 60eqtri 2461 . . . 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 16422 . 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 rngmnd 16652 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
6564adantr 465 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Mnd )
66 prex 4532 . . . . . 6  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V
6766a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V )
6866prid1 3981 . . . . . . . . 9  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
6968, 59eleqtrri 2514 . . . . . . . 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 18022 . . . . . . . 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 1302 . . . . . . 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 474 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  e.  ( Base `  R
) )
7415, 4, 2, 3, 8m2detleiblem2 18432 . . . . . . 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 1302 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) ) )  e.  ( Base `  R
) )
7611, 7rngcl 16656 . . . . . 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 1218 . . . . 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 5689 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )
7978fveq2d 5693 . . . . . . 7  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) )
80 fveq1 5688 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( k `  n )  =  ( { <. 1 ,  1
>. ,  <. 2 ,  2 >. } `  n
) )
8180oveq1d 6104 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) )
8281mpteq2dv 4377 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) )
8382oveq2d 6105 . . . . . . 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 6107 . . . . . 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 16447 . . . . 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 1218 . . . 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 4532 . . . . . 6  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V
8887a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V )
8987prid2 3982 . . . . . . . . 9  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
9089, 59eleqtrri 2514 . . . . . . . 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 18022 . . . . . . . 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 1302 . . . . . . 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 474 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  e.  ( Base `  R
) )
9515, 4, 2, 3, 8m2detleiblem2 18432 . . . . . . 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 1302 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
(mulGrp `  R )  gsumg  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) ) )  e.  ( Base `  R
) )
9711, 7rngcl 16656 . . . . . 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 1218 . . . . 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 5689 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )
10099fveq2d 5693 . . . . . . 7  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) )
101 fveq1 5688 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( k `  n )  =  ( { <. 1 ,  2
>. ,  <. 2 ,  1 >. } `  n
) )
102101oveq1d 6104 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) )
103102mpteq2dv 4377 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) )
104103oveq2d 6105 . . . . . . 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 6107 . . . . . 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 16447 . . . . 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 1218 . . . 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 6107 . . 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 2442 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
110 eqid 2441 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
11115, 4, 5, 6, 110m2detleiblem5 18429 . . . . . . 7  |-  ( ( R  e.  Ring  /\  { <. 1 ,  1 >. ,  <. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  =  ( 1r `  R
) )
112109, 111sylan2 474 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )  =  ( 1r `  R
) )
113 eqidd 2442 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
1148, 7mgpplusg 16593 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
11515, 4, 2, 3, 8, 114m2detleiblem3 18433 . . . . . . 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 1218 . . . . . 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 6107 . . . . 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 3981 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
119118, 15eleqtrri 2514 . . . . . . . . 9  |-  1  e.  N
120119a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  1  e.  N )
1213eleq2i 2505 . . . . . . . . . 10  |-  ( M  e.  B  <->  M  e.  ( Base `  A )
)
122121biimpi 194 . . . . . . . . 9  |-  ( M  e.  B  ->  M  e.  ( Base `  A
) )
123122adantl 466 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  M  e.  ( Base `  A
) )
1242, 11matecl 18324 . . . . . . . 8  |-  ( ( 1  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 1 )  e.  ( Base `  R ) )
125120, 120, 123, 124syl3anc 1218 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 1 )  e.  ( Base `  R
) )
126 prid2g 3980 . . . . . . . . . . 11  |-  ( 2  e.  NN  ->  2  e.  { 1 ,  2 } )
12757, 126ax-mp 5 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
128127, 15eleqtrri 2514 . . . . . . . . 9  |-  2  e.  N
129128a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  2  e.  N )
1302, 11matecl 18324 . . . . . . . 8  |-  ( ( 2  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 2 )  e.  ( Base `  R ) )
131129, 129, 123, 130syl3anc 1218 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 2 )  e.  ( Base `  R
) )
13211, 7rngcl 16656 . . . . . . 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 1218 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )
13411, 7, 110rnglidm 16666 . . . . . 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 470 . . . . 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 2473 . . . 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 2442 . . . . . 6  |-  ( M  e.  B  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
138 eqid 2441 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
13915, 4, 5, 6, 110, 138m2detleiblem6 18430 . . . . . 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 474 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( ZRHom `  R
) `  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
141 eqidd 2442 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
14215, 4, 2, 3, 8, 114m2detleiblem4 18434 . . . . . 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 1218 . . . . 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 6107 . . . 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 6107 . . 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 18324 . . . . . 6  |-  ( ( 2  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 1 )  e.  ( Base `  R ) )
147129, 120, 123, 146syl3anc 1218 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 1 )  e.  ( Base `  R
) )
1482, 11matecl 18324 . . . . . 6  |-  ( ( 1  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 2 )  e.  ( Base `  R ) )
149120, 129, 123, 148syl3anc 1218 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 2 )  e.  ( Base `  R
) )
15011, 7rngcl 16656 . . . . 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 1218 . . . 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 18431 . . . 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 1218 . . 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 2477 . 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 2477 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 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970    u. cun 3324    i^i cin 3325   (/)c0 3635   {csn 3875   {cpr 3877   <.cop 3881    e. cmpt 4348   ` cfv 5416  (class class class)co 6089   Fincfn 7308   1c1 9281   NNcn 10320   2c2 10369   Basecbs 14172   +g cplusg 14236   .rcmulr 14237    gsumg cgsu 14377   Mndcmnd 15407   invgcminusg 15409   -gcsg 15411   SymGrpcsymg 15880  pmSgncpsgn 15993  CMndccmn 16275  mulGrpcmgp 16589   1rcur 16601   Ringcrg 16643   ZRHomczrh 17929   Mat cmat 18278   maDet cmdat 18393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-tpos 6743  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-substr 12231  df-splice 12232  df-reverse 12233  df-s2 12473  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-0g 14378  df-gsum 14379  df-prds 14384  df-pws 14386  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-ghm 15743  df-gim 15785  df-cntz 15833  df-oppg 15859  df-symg 15881  df-pmtr 15946  df-psgn 15995  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-rng 16645  df-cring 16646  df-rnghom 16804  df-subrg 16861  df-sra 17251  df-rgmod 17252  df-cnfld 17817  df-zring 17882  df-zrh 17933  df-dsmm 18155  df-frlm 18170  df-mat 18280  df-mdet 18394
This theorem is referenced by: (None)
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