Definition df-mdet 22807

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22809). 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 22817. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22824, the homogeneity by mdetrsca 22825. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22830) and normalized (see mdet1 22823). Finally, uniqueness is shown by mdetuni 22844. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22809. (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 22806	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3450	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1569	. . . . . 6 class 𝑛
7	3	cv 1569	. . . . . 6 class 𝑟
8		cmat 22629	. . . . . 6 class Mat
9	6, 7, 8	co 7413	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17301	. . . . 5 class Base
11	9, 10	cfv 6533	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19496	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6533	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6533	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1569	. . . . . . . 8 class 𝑝
17		czrh 21712	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6533	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19616	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6533	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5659	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6533	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20273	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6533	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1569	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6533	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1569	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7413	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5186	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17525	. . . . . . . 8 class Σg
32	24, 30, 31	co 7413	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17343	. . . . . . . 8 class .r
34	7, 33	cfv 6533	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7413	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5186	. . . . 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 7413	. . . 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 5186	. . 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 7415	. 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  22808