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 ( 𝜑𝐴𝑉 )
lmat22.b ( 𝜑𝐵𝑉 )
lmat22.c ( 𝜑𝐶𝑉 )
lmat22.d ( 𝜑𝐷𝑉 )
lmat22det.t · = ( .r𝑅 )
lmat22det.s = ( -g𝑅 )
lmat22det.v 𝑉 = ( Base ‘ 𝑅 )
lmat22det.j 𝐽 = ( ( 1 ... 2 ) maDet 𝑅 )
lmat22det.r ( 𝜑𝑅 ∈ Ring )
Assertion lmat22det ( 𝜑 → ( 𝐽𝑀 ) = ( ( 𝐴 · 𝐷 ) ( 𝐶 · 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 lmat22.m 𝑀 = ( litMat ‘ ⟨“ ⟨“ 𝐴 𝐵 ”⟩ ⟨“ 𝐶 𝐷 ”⟩ ”⟩ )
2 lmat22.a ( 𝜑𝐴𝑉 )
3 lmat22.b ( 𝜑𝐵𝑉 )
4 lmat22.c ( 𝜑𝐶𝑉 )
5 lmat22.d ( 𝜑𝐷𝑉 )
6 lmat22det.t · = ( .r𝑅 )
7 lmat22det.s = ( -g𝑅 )
8 lmat22det.v 𝑉 = ( Base ‘ 𝑅 )
9 lmat22det.j 𝐽 = ( ( 1 ... 2 ) maDet 𝑅 )
10 lmat22det.r ( 𝜑𝑅 ∈ Ring )
11 2nn 2 ∈ ℕ
12 11 a1i ( 𝜑 → 2 ∈ ℕ )
13 2 3 s2cld ( 𝜑 → ⟨“ 𝐴 𝐵 ”⟩ ∈ Word 𝑉 )
14 4 5 s2cld ( 𝜑 → ⟨“ 𝐶 𝐷 ”⟩ ∈ Word 𝑉 )
15 13 14 s2cld ( 𝜑 → ⟨“ ⟨“ 𝐴 𝐵 ”⟩ ⟨“ 𝐶 𝐷 ”⟩ ”⟩ ∈ Word Word 𝑉 )
16 s2len ( ♯ ‘ ⟨“ ⟨“ 𝐴 𝐵 ”⟩ ⟨“ 𝐶 𝐷 ”⟩ ”⟩ ) = 2
17 16 a1i ( 𝜑 → ( ♯ ‘ ⟨“ ⟨“ 𝐴 𝐵 ”⟩ ⟨“ 𝐶 𝐷 ”⟩ ”⟩ ) = 2 )
18 1 2 3 4 5 lmat22lem ( ( 𝜑𝑖 ∈ ( 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 ( 𝜑𝑀 ∈ ( Base ‘ ( ( 1 ... 2 ) Mat 𝑅 ) ) )
22 2z 2 ∈ ℤ
23 fzval3 ( 2 ∈ ℤ → ( 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 ∈ Ring ∧ 𝑀 ∈ ( Base ‘ ( ( 1 ... 2 ) Mat 𝑅 ) ) ) → ( 𝐽𝑀 ) = ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) )
30 10 21 29 syl2anc ( 𝜑 → ( 𝐽𝑀 ) = ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) )
31 1 2 3 4 5 lmat22e11 ( 𝜑 → ( 1 𝑀 1 ) = 𝐴 )
32 1 2 3 4 5 lmat22e22 ( 𝜑 → ( 2 𝑀 2 ) = 𝐷 )
33 31 32 oveq12d ( 𝜑 → ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) = ( 𝐴 · 𝐷 ) )
34 1 2 3 4 5 lmat22e21 ( 𝜑 → ( 2 𝑀 1 ) = 𝐶 )
35 1 2 3 4 5 lmat22e12 ( 𝜑 → ( 1 𝑀 2 ) = 𝐵 )
36 34 35 oveq12d ( 𝜑 → ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) = ( 𝐶 · 𝐵 ) )
37 33 36 oveq12d ( 𝜑 → ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) = ( ( 𝐴 · 𝐷 ) ( 𝐶 · 𝐵 ) ) )
38 30 37 eqtrd ( 𝜑 → ( 𝐽𝑀 ) = ( ( 𝐴 · 𝐷 ) ( 𝐶 · 𝐵 ) ) )