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	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	lmat22.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
	lmat22.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
	lmat22.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑉 )
	lmat22det.t	⊢ · = ( .r ‘ 𝑅 )
	lmat22det.s	⊢ − = ( -g ‘ 𝑅 )
	lmat22det.v	⊢ 𝑉 = ( Base ‘ 𝑅 )
	lmat22det.j	⊢ 𝐽 = ( ( 1 ... 2 ) maDet 𝑅 )
	lmat22det.r	⊢ ( 𝜑 → 𝑅 ∈ Ring )
Assertion	lmat22det	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑀 ) = ( ( 𝐴 · 𝐷 ) − ( 𝐶 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lmat22.m	⊢ 𝑀 = ( litMat ‘ ⟨“ ⟨“ 𝐴 𝐵 ”⟩ ⟨“ 𝐶 𝐷 ”⟩ ”⟩ )
2		lmat22.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3		lmat22.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
4		lmat22.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
5		lmat22.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑉 )
6		lmat22det.t	⊢ · = ( .r ‘ 𝑅 )
7		lmat22det.s	⊢ − = ( -g ‘ 𝑅 )
8		lmat22det.v	⊢ 𝑉 = ( Base ‘ 𝑅 )
9		lmat22det.j	⊢ 𝐽 = ( ( 1 ... 2 ) maDet 𝑅 )
10		lmat22det.r	⊢ ( 𝜑 → 𝑅 ∈ 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	⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 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	⊢ ( 𝜑 → 𝑀 ∈ ( 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 ) ) ) → ( 𝐽 ‘ 𝑀 ) = ( ( ( 1 𝑀 1 ) · ( 2 𝑀 2 ) ) − ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) ) )
30	10 21 29	syl2anc	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑀 ) = ( ( ( 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 → ( 𝐽 ‘ 𝑀 ) = ( ( 𝐴 · 𝐷 ) − ( 𝐶 · 𝐵 ) ) )