Definition df-mdet 22742

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22744). 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 22752. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22759, the homogeneity by mdetrsca 22760. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22765) and normalized (see mdet1 22758). Finally, uniqueness is shown by mdetuni 22779. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22744. (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 22741	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3455	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1569	. . . . . 6 class 𝑛
7	3	cv 1569	. . . . . 6 class 𝑟
8		cmat 22564	. . . . . 6 class Mat
9	6, 7, 8	co 7410	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17264	. . . . 5 class Base
11	9, 10	cfv 6536	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19434	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6536	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6536	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1569	. . . . . . . 8 class 𝑝
17		czrh 21649	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6536	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19554	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6536	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5665	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6536	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20211	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6536	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1569	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6536	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1569	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7410	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5192	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17488	. . . . . . . 8 class Σg
32	24, 30, 31	co 7410	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17306	. . . . . . . 8 class .r
34	7, 33	cfv 6536	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7410	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5192	. . . . 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 7410	. . . 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 5192	. . 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 7412	. 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 referenced by:  mdetfval  22743