Definition df-mdet 22541

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22543). 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 22551. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22558, the homogeneity by mdetrsca 22559. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22564) and normalized (see mdet1 22557). Finally, uniqueness is shown by mdetuni 22578. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22543. (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 22540	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3442	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1541	. . . . . 6 class 𝑛
7	3	cv 1541	. . . . . 6 class 𝑟
8		cmat 22363	. . . . . 6 class Mat
9	6, 7, 8	co 7368	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17148	. . . . 5 class Base
11	9, 10	cfv 6500	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19310	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6500	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6500	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1541	. . . . . . . 8 class 𝑝
17		czrh 21466	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6500	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19430	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6500	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5636	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6500	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20087	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6500	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1541	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6500	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1541	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7368	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5181	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17372	. . . . . . . 8 class Σg
32	24, 30, 31	co 7368	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17190	. . . . . . . 8 class .r
34	7, 33	cfv 6500	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7368	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5181	. . . . 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 7368	. . . 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 5181	. . 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 7370	. 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 1542	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  22542