Definition df-mdet 21197

Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 21199). 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 21207. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 21214, the homogeneity by mdetrsca 21215. Furthermore, it is shown that the determinant function is alternating (see mdetralt 21220) and normalized (see mdet1 21213). Finally, the uniqueness is shown by mdetuni 21234. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 21199. (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 21196	. 2 class maDet
2		vn	. . 3 setvar 𝑛
3		vr	. . 3 setvar 𝑟
4		cvv 3497	. . 3 class V
5		vm	. . . 4 setvar 𝑚
6	2	cv 1535	. . . . . 6 class 𝑛
7	3	cv 1535	. . . . . 6 class 𝑟
8		cmat 21019	. . . . . 6 class Mat
9	6, 7, 8	co 7159	. . . . 5 class (𝑛 Mat 𝑟)
10		cbs 16486	. . . . 5 class Base
11	9, 10	cfv 6358	. . . 4 class (Base‘(𝑛 Mat 𝑟))
12		vp	. . . . . 6 setvar 𝑝
13		csymg 18498	. . . . . . . 8 class SymGrp
14	6, 13	cfv 6358	. . . . . . 7 class (SymGrp‘𝑛)
15	14, 10	cfv 6358	. . . . . 6 class (Base‘(SymGrp‘𝑛))
16	12	cv 1535	. . . . . . . 8 class 𝑝
17		czrh 20650	. . . . . . . . . 10 class ℤRHom
18	7, 17	cfv 6358	. . . . . . . . 9 class (ℤRHom‘𝑟)
19		cpsgn 18620	. . . . . . . . . 10 class pmSgn
20	6, 19	cfv 6358	. . . . . . . . 9 class (pmSgn‘𝑛)
21	18, 20	ccom 5562	. . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
22	16, 21	cfv 6358	. . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23		cmgp 19242	. . . . . . . . 9 class mulGrp
24	7, 23	cfv 6358	. . . . . . . 8 class (mulGrp‘𝑟)
25		vx	. . . . . . . . 9 setvar 𝑥
26	25	cv 1535	. . . . . . . . . . 11 class 𝑥
27	26, 16	cfv 6358	. . . . . . . . . 10 class (𝑝‘𝑥)
28	5	cv 1535	. . . . . . . . . 10 class 𝑚
29	27, 26, 28	co 7159	. . . . . . . . 9 class ((𝑝‘𝑥)𝑚𝑥)
30	25, 6, 29	cmpt 5149	. . . . . . . 8 class (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))
31		cgsu 16717	. . . . . . . 8 class Σg
32	24, 30, 31	co 7159	. . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))
33		cmulr 16569	. . . . . . . 8 class .r
34	7, 33	cfv 6358	. . . . . . 7 class (.r‘𝑟)
35	22, 32, 34	co 7159	. . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥))))
36	12, 15, 35	cmpt 5149	. . . . 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 7159	. . . 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 5149	. . 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 7161	. 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 1536	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  21198