Definition df-mdet 22955

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22957). 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 22965. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22972, the homogeneity by mdetrsca 22973. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22978) and normalized (see mdet1 22971). Finally, uniqueness is shown by mdetuni 22992. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22957. (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 22954	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3454	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1569	. . . . . 6 class 𝑛
7	3	cv 1569	. . . . . 6 class 𝑟
8		cmat 22777	. . . . . 6 class Mat
9	6, 7, 8	co 7451	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17431	. . . . 5 class Base
11	9, 10	cfv 6566	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19634	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6566	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6566	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1569	. . . . . . . 8 class 𝑝
17		czrh 21860	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6566	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19754	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6566	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5662	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6566	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20411	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6566	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1569	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6566	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1569	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7451	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5192	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17661	. . . . . . . 8 class Σg
32	24, 30, 31	co 7451	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17473	. . . . . . . 8 class .r
34	7, 33	cfv 6566	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7451	. . . . . 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 7451	. . . 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 7453	. 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  22956