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Definition df-mdet 22086
Description: Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22088). 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 22096. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22103, the homogeneity by mdetrsca 22104. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22109) and normalized (see mdet1 22102). Finally, uniqueness is shown by mdetuni 22123. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22088. (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 22085 . 2 class maDet
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cvv 3474 . . 3 class V
5 vm . . . 4 setvar 𝑚
62cv 1540 . . . . . 6 class 𝑛
73cv 1540 . . . . . 6 class 𝑟
8 cmat 21906 . . . . . 6 class Mat
96, 7, 8co 7408 . . . . 5 class (𝑛 Mat 𝑟)
10 cbs 17143 . . . . 5 class Base
119, 10cfv 6543 . . . 4 class (Base‘(𝑛 Mat 𝑟))
12 vp . . . . . 6 setvar 𝑝
13 csymg 19233 . . . . . . . 8 class SymGrp
146, 13cfv 6543 . . . . . . 7 class (SymGrp‘𝑛)
1514, 10cfv 6543 . . . . . 6 class (Base‘(SymGrp‘𝑛))
1612cv 1540 . . . . . . . 8 class 𝑝
17 czrh 21048 . . . . . . . . . 10 class ℤRHom
187, 17cfv 6543 . . . . . . . . 9 class (ℤRHom‘𝑟)
19 cpsgn 19356 . . . . . . . . . 10 class pmSgn
206, 19cfv 6543 . . . . . . . . 9 class (pmSgn‘𝑛)
2118, 20ccom 5680 . . . . . . . 8 class ((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))
2216, 21cfv 6543 . . . . . . 7 class (((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)
23 cmgp 19986 . . . . . . . . 9 class mulGrp
247, 23cfv 6543 . . . . . . . 8 class (mulGrp‘𝑟)
25 vx . . . . . . . . 9 setvar 𝑥
2625cv 1540 . . . . . . . . . . 11 class 𝑥
2726, 16cfv 6543 . . . . . . . . . 10 class (𝑝𝑥)
285cv 1540 . . . . . . . . . 10 class 𝑚
2927, 26, 28co 7408 . . . . . . . . 9 class ((𝑝𝑥)𝑚𝑥)
3025, 6, 29cmpt 5231 . . . . . . . 8 class (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))
31 cgsu 17385 . . . . . . . 8 class Σg
3224, 30, 31co 7408 . . . . . . 7 class ((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))
33 cmulr 17197 . . . . . . . 8 class .r
347, 33cfv 6543 . . . . . . 7 class (.r𝑟)
3522, 32, 34co 7408 . . . . . 6 class ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))
3612, 15, 35cmpt 5231 . . . . 5 class (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥)))))
377, 36, 31co 7408 . . . 4 class (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))
385, 11, 37cmpt 5231 . . 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 7410 . 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 1541 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  22087
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