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Theorem m2detleib 19110
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 2443 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2443 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2443 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 m2detleib.t . . . 4  |-  .x.  =  ( .r `  R )
8 eqid 2443 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib1 19070 . . 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 2443 . . 3  |-  ( Base `  R )  =  (
Base `  R )
12 eqid 2443 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
13 ringcmn 17207 . . . 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 7797 . . . . . 6  |-  { 1 ,  2 }  e.  Fin
1715, 16eqeltri 2527 . . . . 5  |-  N  e. 
Fin
18 eqid 2443 . . . . . 6  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
1918, 4symgbasfi 16389 . . . . 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 18607 . . . . . 6  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2517, 24mp3an2 1313 . . . . 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 19107 . . . . 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 1229 . . . 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 17190 . . . 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 1229 . . 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 4701 . . . . . . . 8  |-  <. 1 ,  1 >.  e.  _V
35 opex 4701 . . . . . . . 8  |-  <. 2 ,  2 >.  e.  _V
3634, 35pm3.2i 455 . . . . . . 7  |-  ( <.
1 ,  1 >.  e.  _V  /\  <. 2 ,  2 >.  e.  _V )
37 opex 4701 . . . . . . . 8  |-  <. 1 ,  2 >.  e.  _V
38 opex 4701 . . . . . . . 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 10755 . . . . . . . . . 10  |-  1  =/=  2
4241olci 391 . . . . . . . . 9  |-  ( 1  =/=  1  \/  1  =/=  2 )
43 1ex 9594 . . . . . . . . . 10  |-  1  e.  _V
4443, 43opthne 4717 . . . . . . . . 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 4717 . . . . . . . . 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 4196 . . . . 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 4076 . . . 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 10700 . . . . . 6  |-  2  e.  NN
5818, 4, 15symg2bas 16401 . . . . . 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 4017 . . . . 5  |-  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  =  ( { { <. 1 ,  1 >. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } } )
6159, 60eqtri 2472 . . . 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 16928 . 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 17185 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
6564adantr 465 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Mnd )
66 prex 4679 . . . . . 6  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V
6766a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V )
6866prid1 4123 . . . . . . . . 9  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
6968, 59eleqtrri 2530 . . . . . . . 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 18607 . . . . . . . 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 1313 . . . . . . 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 19107 . . . . . . 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 1313 . . . . . 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 17190 . . . . . 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 1229 . . . . 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 5856 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )
7978fveq2d 5860 . . . . . . 7  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) )
80 fveq1 5855 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( k `  n )  =  ( { <. 1 ,  1
>. ,  <. 2 ,  2 >. } `  n
) )
8180oveq1d 6296 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) )
8281mpteq2dv 4524 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) )
8382oveq2d 6297 . . . . . . 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 6299 . . . . . 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 16959 . . . . 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 1229 . . . 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 4679 . . . . . 6  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V
8887a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V )
8987prid2 4124 . . . . . . . . 9  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
9089, 59eleqtrri 2530 . . . . . . . 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 18607 . . . . . . . 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 1313 . . . . . . 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 19107 . . . . . . 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 1313 . . . . . 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 17190 . . . . . 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 1229 . . . . 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 5856 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )
10099fveq2d 5860 . . . . . . 7  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) )
101 fveq1 5855 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( k `  n )  =  ( { <. 1 ,  2
>. ,  <. 2 ,  1 >. } `  n
) )
102101oveq1d 6296 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) )
103102mpteq2dv 4524 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) )
104103oveq2d 6297 . . . . . . 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 6299 . . . . . 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 16959 . . . . 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 1229 . . . 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 6299 . . 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 2444 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
110 eqid 2443 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
11115, 4, 5, 6, 110m2detleiblem5 19104 . . . . . . 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 2444 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
1148, 7mgpplusg 17123 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
11515, 4, 2, 3, 8, 114m2detleiblem3 19108 . . . . . . 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 1229 . . . . . 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 6299 . . . . 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 4123 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
119118, 15eleqtrri 2530 . . . . . . . . 9  |-  1  e.  N
120119a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  1  e.  N )
1213eleq2i 2521 . . . . . . . . . 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 18904 . . . . . . . 8  |-  ( ( 1  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 1 )  e.  ( Base `  R ) )
125120, 120, 123, 124syl3anc 1229 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 1 )  e.  ( Base `  R
) )
126 prid2g 4122 . . . . . . . . . . 11  |-  ( 2  e.  NN  ->  2  e.  { 1 ,  2 } )
12757, 126ax-mp 5 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
128127, 15eleqtrri 2530 . . . . . . . . 9  |-  2  e.  N
129128a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  2  e.  N )
1302, 11matecl 18904 . . . . . . . 8  |-  ( ( 2  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 2 )  e.  ( Base `  R ) )
131129, 129, 123, 130syl3anc 1229 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 2 )  e.  ( Base `  R
) )
13211, 7ringcl 17190 . . . . . . 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 1229 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )
13411, 7, 110ringlidm 17200 . . . . . 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 2484 . . . 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 2444 . . . . . 6  |-  ( M  e.  B  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
138 eqid 2443 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
13915, 4, 5, 6, 110, 138m2detleiblem6 19105 . . . . . 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 2444 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
14215, 4, 2, 3, 8, 114m2detleiblem4 19109 . . . . . 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 1229 . . . . 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 6299 . . . 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 6299 . . 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 18904 . . . . . 6  |-  ( ( 2  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 1 )  e.  ( Base `  R ) )
147129, 120, 123, 146syl3anc 1229 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 1 )  e.  ( Base `  R
) )
1482, 11matecl 18904 . . . . . 6  |-  ( ( 1  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 2 )  e.  ( Base `  R ) )
149120, 129, 123, 148syl3anc 1229 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 2 )  e.  ( Base `  R
) )
15011, 7ringcl 17190 . . . . 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 1229 . . . 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 19106 . . . 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 1229 . . 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 2488 . 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 2488 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 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    u. cun 3459    i^i cin 3460   (/)c0 3770   {csn 4014   {cpr 4016   <.cop 4020    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   Fincfn 7518   1c1 9496   NNcn 10543   2c2 10592   Basecbs 14613   +g cplusg 14678   .rcmulr 14679    gsumg cgsu 14819   Mndcmnd 15897   invgcminusg 16032   -gcsg 16033   SymGrpcsymg 16380  pmSgncpsgn 16492  CMndccmn 16776  mulGrpcmgp 17119   1rcur 17131   Ringcrg 17176   ZRHomczrh 18514   Mat cmat 18886   maDet cmdat 19063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-xor 1365  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-rp 11231  df-fz 11683  df-fzo 11806  df-seq 12089  df-exp 12148  df-fac 12335  df-bc 12362  df-hash 12387  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12794  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-hom 14702  df-cco 14703  df-0g 14820  df-gsum 14821  df-prds 14826  df-pws 14828  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-mhm 15944  df-submnd 15945  df-grp 16035  df-minusg 16036  df-sbg 16037  df-mulg 16038  df-subg 16176  df-ghm 16243  df-gim 16285  df-cntz 16333  df-oppg 16359  df-symg 16381  df-pmtr 16445  df-psgn 16494  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-ring 17178  df-cring 17179  df-rnghom 17342  df-subrg 17405  df-sra 17796  df-rgmod 17797  df-cnfld 18399  df-zring 18467  df-zrh 18518  df-dsmm 18740  df-frlm 18755  df-mat 18887  df-mdet 19064
This theorem is referenced by: (None)
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