Definition df-mdet 22307

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22309). 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 22317. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22324, the homogeneity by mdetrsca 22325. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22330) and normalized (see mdet1 22323). Finally, uniqueness is shown by mdetuni 22344. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22309. (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 22306	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3472	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1538	. . . . . 6 class 𝑛
7	3	cv 1538	. . . . . 6 class 𝑟
8		cmat 22127	. . . . . 6 class Mat
9	6, 7, 8	co 7411	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 17148	. . . . 5 class Base
11	9, 10	cfv 6542	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 19275	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6542	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6542	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1538	. . . . . . . 8 class 𝑝
17		czrh 21268	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6542	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 19398	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6542	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5679	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6542	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 20028	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6542	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1538	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6542	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1538	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7411	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5230	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 17390	. . . . . . . 8 class Σg
32	24, 30, 31	co 7411	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 17202	. . . . . . . 8 class .r
34	7, 33	cfv 6542	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7411	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5230	. . . . 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 7411	. . . 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 5230	. . 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 7413	. 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 1539	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  22308