Definition df-mdet 22813

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22815). The properties of the axiomatic definition of a determinant according to [Weierstrass] p. 272 are derived from this definition as theorems: "The determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring". Functionality is shown by mdetf 22823. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22830, the homogeneity by mdetrsca 22831. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22836) and normalized (see mdet1 22829). Finally, uniqueness is shown by mdetuni 22850. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22815. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 10-Jul-2018.)

Assertion

Ref	Expression
df-mdet	⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))

Distinct variable group:   𝑛,𝑟,𝑚,𝑝,𝑥

Detailed syntax breakdown of Definition df-mdet

Step	Hyp	Ref	Expression
1		cmdat 22812	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3453	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1569	. . . . . 6 class 𝑛
7	3	cv 1569	. . . . . 6 class 𝑟
8		cmat 22635	. . . . . 6 class Mat
9	6, 7, 8	co 7417	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17307	. . . . 5 class Base
11	9, 10	cfv 6537	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19502	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6537	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6537	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1569	. . . . . . . 8 class 𝑝
17		czrh 21718	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6537	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19622	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6537	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5663	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6537	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20279	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6537	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1569	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6537	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1569	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7417	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5190	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17531	. . . . . . . 8 class Σg
32	24, 30, 31	co 7417	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17349	. . . . . . . 8 class .r
34	7, 33	cfv 6537	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7417	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5190	. . . . 5 class (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))))
37	7, 36, 31	co 7417	. . . 4 class (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))
38	5, 11, 37	cmpt 5190	. . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))
39	2, 3, 4, 4, 38	cmpo 7419	. 2 class (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))
40	1, 39	wceq 1570	1 wff maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))))))

This definition is used by:  mdetfval  22814