Definition df-mdet 21196

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 21198). 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". The functionality is shown by mdetf 21206. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 21213, the homogeneity by mdetrsca 21214. Furthermore, it is shown that the determinant function is alternating (see mdetralt 21219) and normalized (see mdet1 21212). Finally, the uniqueness is shown by mdetuni 21233. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 21198. (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 21195	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3496	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1536	. . . . . 6 class 𝑛
7	3	cv 1536	. . . . . 6 class 𝑟
8		cmat 21018	. . . . . 6 class Mat
9	6, 7, 8	co 7158	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 16485	. . . . 5 class Base
11	9, 10	cfv 6357	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 18497	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6357	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6357	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1536	. . . . . . . 8 class 𝑝
17		czrh 20649	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6357	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 18619	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6357	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5561	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6357	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 19241	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6357	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1536	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6357	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1536	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7158	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5148	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 16716	. . . . . . . 8 class Σg
32	24, 30, 31	co 7158	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 16568	. . . . . . . 8 class .r
34	7, 33	cfv 6357	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7158	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5148	. . . . 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 7158	. . . 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 5148	. . 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 7160	. 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 1537	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  21197