Definition df-mdet 22568

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22570). 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 22578. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22585, the homogeneity by mdetrsca 22586. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22591) and normalized (see mdet1 22584). Finally, uniqueness is shown by mdetuni 22605. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22570. (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 22567	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3431	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1546	. . . . . 6 class 𝑛
7	3	cv 1546	. . . . . 6 class 𝑟
8		cmat 22390	. . . . . 6 class Mat
9	6, 7, 8	co 7356	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17170	. . . . 5 class Base
11	9, 10	cfv 6485	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19335	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6485	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6485	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1546	. . . . . . . 8 class 𝑝
17		czrh 21474	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6485	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19455	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6485	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5622	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6485	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20112	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6485	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1546	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6485	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1546	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7356	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5153	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17394	. . . . . . . 8 class Σg
32	24, 30, 31	co 7356	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17212	. . . . . . . 8 class .r
34	7, 33	cfv 6485	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7356	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5153	. . . . 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 7356	. . . 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 5153	. . 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 7358	. 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 1547	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 referenced by:  mdetfval  22569