MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  m2detleib Structured version   Unicode version

Theorem m2detleib 18893
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 2460 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
5 eqid 2460 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
6 eqid 2460 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
7 m2detleib.t . . . 4  |-  .x.  =  ( .r `  R )
8 eqid 2460 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
91, 2, 3, 4, 5, 6, 7, 8mdetleib1 18853 . . 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 2460 . . 3  |-  ( Base `  R )  =  (
Base `  R )
12 eqid 2460 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
13 rngcmn 17009 . . . 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 7784 . . . . . 6  |-  { 1 ,  2 }  e.  Fin
1715, 16eqeltri 2544 . . . . 5  |-  N  e. 
Fin
18 eqid 2460 . . . . . 6  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
1918, 4symgbasfi 16199 . . . . 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 18390 . . . . . 6  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  k  e.  ( Base `  ( SymGrp `
 N ) ) )  ->  ( ( ZRHom `  R ) `  ( (pmSgn `  N ) `  k ) )  e.  ( Base `  R
) )
2517, 24mp3an2 1307 . . . . 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 18890 . . . . 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 1223 . . . 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 16992 . . . 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 1223 . . 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 4704 . . . . . . . 8  |-  <. 1 ,  1 >.  e.  _V
35 opex 4704 . . . . . . . 8  |-  <. 2 ,  2 >.  e.  _V
3634, 35pm3.2i 455 . . . . . . 7  |-  ( <.
1 ,  1 >.  e.  _V  /\  <. 2 ,  2 >.  e.  _V )
37 opex 4704 . . . . . . . 8  |-  <. 1 ,  2 >.  e.  _V
38 opex 4704 . . . . . . . 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 10737 . . . . . . . . . 10  |-  1  =/=  2
4241olci 391 . . . . . . . . 9  |-  ( 1  =/=  1  \/  1  =/=  2 )
43 1ex 9580 . . . . . . . . . 10  |-  1  e.  _V
4443, 43opthne 4720 . . . . . . . . 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 4720 . . . . . . . . 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 4200 . . . . 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 4082 . . . 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 10682 . . . . . 6  |-  2  e.  NN
5818, 4, 15symg2bas 16211 . . . . . 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 4023 . . . . 5  |-  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }  =  ( { { <. 1 ,  1 >. ,  <. 2 ,  2 >. } }  u.  { { <. 1 ,  2 >. ,  <. 2 ,  1
>. } } )
6159, 60eqtri 2489 . . . 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 16734 . 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 16988 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
6564adantr 465 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  R  e.  Mnd )
66 prex 4682 . . . . . 6  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V
6766a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  _V )
6866prid1 4128 . . . . . . . . 9  |-  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
6968, 59eleqtrri 2547 . . . . . . . 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 18390 . . . . . . . 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 1307 . . . . . . 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 18890 . . . . . . 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 1307 . . . . . 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 16992 . . . . . 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 1223 . . . . 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 5857 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ) )
7978fveq2d 5861 . . . . . . 7  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  1
>. ,  <. 2 ,  2 >. } ) ) )
80 fveq1 5856 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( k `  n )  =  ( { <. 1 ,  1
>. ,  <. 2 ,  2 >. } `  n
) )
8180oveq1d 6290 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  1 >. ,  <. 2 ,  2 >. } `
 n ) M n ) )
8281mpteq2dv 4527 . . . . . . . 8  |-  ( k  =  { <. 1 ,  1 >. ,  <. 2 ,  2 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  1 >. , 
<. 2 ,  2
>. } `  n ) M n ) ) )
8382oveq2d 6291 . . . . . . 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 6293 . . . . . 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 16765 . . . . 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 1223 . . . 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 4682 . . . . . 6  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V
8887a1i 11 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  _V )
8987prid2 4129 . . . . . . . . 9  |-  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  e.  { { <. 1 ,  1 >. ,  <. 2 ,  2
>. } ,  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } }
9089, 59eleqtrri 2547 . . . . . . . 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 18390 . . . . . . . 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 1307 . . . . . . 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 18890 . . . . . . 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 1307 . . . . . 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 16992 . . . . . 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 1223 . . . . 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 5857 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( (pmSgn `  N ) `  k
)  =  ( (pmSgn `  N ) `  { <. 1 ,  2 >. ,  <. 2 ,  1
>. } ) )
10099fveq2d 5861 . . . . . . 7  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  k ) )  =  ( ( ZRHom `  R ) `  (
(pmSgn `  N ) `  { <. 1 ,  2
>. ,  <. 2 ,  1 >. } ) ) )
101 fveq1 5856 . . . . . . . . . 10  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( k `  n )  =  ( { <. 1 ,  2
>. ,  <. 2 ,  1 >. } `  n
) )
102101oveq1d 6290 . . . . . . . . 9  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( ( k `
 n ) M n )  =  ( ( { <. 1 ,  2 >. ,  <. 2 ,  1 >. } `
 n ) M n ) )
103102mpteq2dv 4527 . . . . . . . 8  |-  ( k  =  { <. 1 ,  2 >. ,  <. 2 ,  1 >. }  ->  ( n  e.  N  |->  ( ( k `
 n ) M n ) )  =  ( n  e.  N  |->  ( ( { <. 1 ,  2 >. , 
<. 2 ,  1
>. } `  n ) M n ) ) )
104103oveq2d 6291 . . . . . . 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 6293 . . . . . 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 16765 . . . . 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 1223 . . . 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 6293 . . 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 2461 . . . . . . 7  |-  ( M  e.  B  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
110 eqid 2460 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
11115, 4, 5, 6, 110m2detleiblem5 18887 . . . . . . 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 2461 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  1 >. , 
<. 2 ,  2
>. }  =  { <. 1 ,  1 >. , 
<. 2 ,  2
>. } )
1148, 7mgpplusg 16928 . . . . . . . 8  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
11515, 4, 2, 3, 8, 114m2detleiblem3 18891 . . . . . . 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 1223 . . . . . 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 6293 . . . . 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 4128 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
119118, 15eleqtrri 2547 . . . . . . . . 9  |-  1  e.  N
120119a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  1  e.  N )
1213eleq2i 2538 . . . . . . . . . 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 18687 . . . . . . . 8  |-  ( ( 1  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 1 )  e.  ( Base `  R ) )
125120, 120, 123, 124syl3anc 1223 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 1 )  e.  ( Base `  R
) )
126 prid2g 4127 . . . . . . . . . . 11  |-  ( 2  e.  NN  ->  2  e.  { 1 ,  2 } )
12757, 126ax-mp 5 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
128127, 15eleqtrri 2547 . . . . . . . . 9  |-  2  e.  N
129128a1i 11 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  2  e.  N )
1302, 11matecl 18687 . . . . . . . 8  |-  ( ( 2  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 2 )  e.  ( Base `  R ) )
131129, 129, 123, 130syl3anc 1223 . . . . . . 7  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 2 )  e.  ( Base `  R
) )
13211, 7rngcl 16992 . . . . . . 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 1223 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( 1 M 1 )  .x.  ( 2 M 2 ) )  e.  ( Base `  R
) )
13411, 7, 110rnglidm 17002 . . . . . 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 2501 . . . 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 2461 . . . . . 6  |-  ( M  e.  B  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
138 eqid 2460 . . . . . . 7  |-  ( invg `  R )  =  ( invg `  R )
13915, 4, 5, 6, 110, 138m2detleiblem6 18888 . . . . . 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 2461 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  { <. 1 ,  2 >. , 
<. 2 ,  1
>. }  =  { <. 1 ,  2 >. , 
<. 2 ,  1
>. } )
14215, 4, 2, 3, 8, 114m2detleiblem4 18892 . . . . . 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 1223 . . . . 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 6293 . . . 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 6293 . . 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 18687 . . . . . 6  |-  ( ( 2  e.  N  /\  1  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 2 M 1 )  e.  ( Base `  R ) )
147129, 120, 123, 146syl3anc 1223 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
2 M 1 )  e.  ( Base `  R
) )
1482, 11matecl 18687 . . . . . 6  |-  ( ( 1  e.  N  /\  2  e.  N  /\  M  e.  ( Base `  A ) )  -> 
( 1 M 2 )  e.  ( Base `  R ) )
149120, 129, 123, 148syl3anc 1223 . . . . 5  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
1 M 2 )  e.  ( Base `  R
) )
15011, 7rngcl 16992 . . . . 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 1223 . . . 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 18889 . . . 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 1223 . . 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 2505 . 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 2505 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 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106    u. cun 3467    i^i cin 3468   (/)c0 3778   {csn 4020   {cpr 4022   <.cop 4026    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Fincfn 7506   1c1 9482   NNcn 10525   2c2 10574   Basecbs 14479   +g cplusg 14544   .rcmulr 14545    gsumg cgsu 14685   Mndcmnd 15715   invgcminusg 15717   -gcsg 15719   SymGrpcsymg 16190  pmSgncpsgn 16303  CMndccmn 16587  mulGrpcmgp 16924   1rcur 16936   Ringcrg 16979   ZRHomczrh 18297   Mat cmat 18669   maDet cmdat 18846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-xor 1356  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-substr 12499  df-splice 12500  df-reverse 12501  df-s2 12763  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-0g 14686  df-gsum 14687  df-prds 14692  df-pws 14694  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-ghm 16053  df-gim 16095  df-cntz 16143  df-oppg 16169  df-symg 16191  df-pmtr 16256  df-psgn 16305  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-rnghom 17141  df-subrg 17203  df-sra 17594  df-rgmod 17595  df-cnfld 18185  df-zring 18250  df-zrh 18301  df-dsmm 18523  df-frlm 18538  df-mat 18670  df-mdet 18847
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator