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Definition df-mdet 21285
 Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 21287). 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 21295. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 21302, the homogeneity by mdetrsca 21303. Furthermore, it is shown that the determinant function is alternating (see mdetralt 21308) and normalized (see mdet1 21301). Finally, uniqueness is shown by mdetuni 21322. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 21287. (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)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))
Distinct variable group:   𝑛,𝑟,𝑚,𝑝,𝑥

Detailed syntax breakdown of Definition df-mdet
StepHypRef Expression
1 cmdat 21284 . 2 class maDet
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cvv 3409 . . 3 class V
5 vm . . . 4 setvar 𝑚
62cv 1537 . . . . . 6 class 𝑛
73cv 1537 . . . . . 6 class 𝑟
8 cmat 21107 . . . . . 6 class Mat
96, 7, 8co 7150 . . . . 5 class (𝑛 Mat 𝑟)
10 cbs 16541 . . . . 5 class Base
119, 10cfv 6335 . . . 4 class (Base‘(𝑛 Mat 𝑟))
12 vp . . . . . 6 setvar 𝑝
13 csymg 18562 . . . . . . . 8 class SymGrp
146, 13cfv 6335 . . . . . . 7 class (SymGrp‘𝑛)
1514, 10cfv 6335 . . . . . 6 class (Base‘(SymGrp‘𝑛))
1612cv 1537 . . . . . . . 8 class 𝑝
17 czrh 20269 . . . . . . . . . 10 class ℤRHom
187, 17cfv 6335 . . . . . . . . 9 class (ℤRHom‘𝑟)
19 cpsgn 18684 . . . . . . . . . 10 class pmSgn
206, 19cfv 6335 . . . . . . . . 9 class (pmSgn‘𝑛)
2118, 20ccom 5528 . . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
2216, 21cfv 6335 . . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23 cmgp 19307 . . . . . . . . 9 class mulGrp
247, 23cfv 6335 . . . . . . . 8 class (mulGrp‘𝑟)
25 vx . . . . . . . . 9 setvar 𝑥
2625cv 1537 . . . . . . . . . . 11 class 𝑥
2726, 16cfv 6335 . . . . . . . . . 10 class (𝑝𝑥)
285cv 1537 . . . . . . . . . 10 class 𝑚
2927, 26, 28co 7150 . . . . . . . . 9 class ((𝑝𝑥)𝑚𝑥)
3025, 6, 29cmpt 5112 . . . . . . . 8 class (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))
31 cgsu 16772 . . . . . . . 8 class Σg
3224, 30, 31co 7150 . . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))
33 cmulr 16624 . . . . . . . 8 class .r
347, 33cfv 6335 . . . . . . 7 class (.r𝑟)
3522, 32, 34co 7150 . . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))
3612, 15, 35cmpt 5112 . . . . 5 class (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))))
377, 36, 31co 7150 . . . 4 class (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))
385, 11, 37cmpt 5112 . . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))))))
392, 3, 4, 4, 38cmpo 7152 . 2 class (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))
401, 39wceq 1538 1 wff maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))
 Colors of variables: wff setvar class This definition is referenced by:  mdetfval  21286
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