Metamath Proof Explorer

Theorem lmat22det

Description: The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020)

Ref Expression
Hypotheses lmat22.m 
|- M = ( litMat ` <" <" A B "> <" C D "> "> )
lmat22.a 
|- ( ph -> A e. V )
lmat22.b 
|- ( ph -> B e. V )
lmat22.c 
|- ( ph -> C e. V )
lmat22.d 
|- ( ph -> D e. V )
lmat22det.t 
|- .x. = ( .r ` R )
lmat22det.s 
|- .- = ( -g ` R )
lmat22det.v 
|- V = ( Base ` R )
lmat22det.j 
|- J = ( ( 1 ... 2 ) maDet R )
lmat22det.r 
|- ( ph -> R e. Ring )
Assertion lmat22det 
|- ( ph -> ( J ` M ) = ( ( A .x. D ) .- ( C .x. B ) ) )

Proof

Step Hyp Ref Expression
1 lmat22.m 
 |-  M = ( litMat ` <" <" A B "> <" C D "> "> )
2 lmat22.a 
 |-  ( ph -> A e. V )
3 lmat22.b 
 |-  ( ph -> B e. V )
4 lmat22.c 
 |-  ( ph -> C e. V )
5 lmat22.d 
 |-  ( ph -> D e. V )
6 lmat22det.t 
 |-  .x. = ( .r ` R )
7 lmat22det.s 
 |-  .- = ( -g ` R )
8 lmat22det.v 
 |-  V = ( Base ` R )
9 lmat22det.j 
 |-  J = ( ( 1 ... 2 ) maDet R )
10 lmat22det.r 
 |-  ( ph -> R e. Ring )
11 2nn 
 |-  2 e. NN
12 11 a1i 
 |-  ( ph -> 2 e. NN )
13 2 3 s2cld 
 |-  ( ph -> <" A B "> e. Word V )
14 4 5 s2cld 
 |-  ( ph -> <" C D "> e. Word V )
15 13 14 s2cld 
 |-  ( ph -> <" <" A B "> <" C D "> "> e. Word Word V )
16 s2len 
 |-  ( # ` <" <" A B "> <" C D "> "> ) = 2
17 16 a1i 
 |-  ( ph -> ( # ` <" <" A B "> <" C D "> "> ) = 2 )
18 1 2 3 4 5 lmat22lem 
 |-  ( ( ph /\ i e. ( 0 ..^ 2 ) ) -> ( # ` ( <" <" A B "> <" C D "> "> ` i ) ) = 2 )
19 eqid 
 |-  ( ( 1 ... 2 ) Mat R ) = ( ( 1 ... 2 ) Mat R )
20 eqid 
 |-  ( Base ` ( ( 1 ... 2 ) Mat R ) ) = ( Base ` ( ( 1 ... 2 ) Mat R ) )
21 1 12 15 17 18 8 19 20 10 lmatcl 
 |-  ( ph -> M e. ( Base ` ( ( 1 ... 2 ) Mat R ) ) )
22 2z 
 |-  2 e. ZZ
23 fzval3 
 |-  ( 2 e. ZZ -> ( 1 ... 2 ) = ( 1 ..^ ( 2 + 1 ) ) )
24 22 23 ax-mp 
 |-  ( 1 ... 2 ) = ( 1 ..^ ( 2 + 1 ) )
25 2p1e3 
 |-  ( 2 + 1 ) = 3
26 25 oveq2i 
 |-  ( 1 ..^ ( 2 + 1 ) ) = ( 1 ..^ 3 )
27 fzo13pr 
 |-  ( 1 ..^ 3 ) = { 1 , 2 }
28 24 26 27 3eqtri 
 |-  ( 1 ... 2 ) = { 1 , 2 }
29 28 9 19 20 7 6 m2detleib 
 |-  ( ( R e. Ring /\ M e. ( Base ` ( ( 1 ... 2 ) Mat R ) ) ) -> ( J ` M ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )
30 10 21 29 syl2anc 
 |-  ( ph -> ( J ` M ) = ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) )
31 1 2 3 4 5 lmat22e11 
 |-  ( ph -> ( 1 M 1 ) = A )
32 1 2 3 4 5 lmat22e22 
 |-  ( ph -> ( 2 M 2 ) = D )
33 31 32 oveq12d 
 |-  ( ph -> ( ( 1 M 1 ) .x. ( 2 M 2 ) ) = ( A .x. D ) )
34 1 2 3 4 5 lmat22e21 
 |-  ( ph -> ( 2 M 1 ) = C )
35 1 2 3 4 5 lmat22e12 
 |-  ( ph -> ( 1 M 2 ) = B )
36 34 35 oveq12d 
 |-  ( ph -> ( ( 2 M 1 ) .x. ( 1 M 2 ) ) = ( C .x. B ) )
37 33 36 oveq12d 
 |-  ( ph -> ( ( ( 1 M 1 ) .x. ( 2 M 2 ) ) .- ( ( 2 M 1 ) .x. ( 1 M 2 ) ) ) = ( ( A .x. D ) .- ( C .x. B ) ) )
38 30 37 eqtrd 
 |-  ( ph -> ( J ` M ) = ( ( A .x. D ) .- ( C .x. B ) ) )