Metamath Proof Explorer


Theorem mdetleib1

Description: Full substitution of an alternative determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015) (Revised by AV, 26-Dec-2018)

Ref Expression
Hypotheses mdetfval1.d D = N maDet R
mdetfval1.a A = N Mat R
mdetfval1.b B = Base A
mdetfval1.p P = Base SymGrp N
mdetfval1.y Y = ℤRHom R
mdetfval1.s S = pmSgn N
mdetfval1.t · ˙ = R
mdetfval1.u U = mulGrp R
Assertion mdetleib1 M B D M = R p P Y S p · ˙ U x N p x M x

Proof

Step Hyp Ref Expression
1 mdetfval1.d D = N maDet R
2 mdetfval1.a A = N Mat R
3 mdetfval1.b B = Base A
4 mdetfval1.p P = Base SymGrp N
5 mdetfval1.y Y = ℤRHom R
6 mdetfval1.s S = pmSgn N
7 mdetfval1.t · ˙ = R
8 mdetfval1.u U = mulGrp R
9 oveq m = M p x m x = p x M x
10 9 mpteq2dv m = M x N p x m x = x N p x M x
11 10 oveq2d m = M U x N p x m x = U x N p x M x
12 11 oveq2d m = M Y S p · ˙ U x N p x m x = Y S p · ˙ U x N p x M x
13 12 mpteq2dv m = M p P Y S p · ˙ U x N p x m x = p P Y S p · ˙ U x N p x M x
14 13 oveq2d m = M R p P Y S p · ˙ U x N p x m x = R p P Y S p · ˙ U x N p x M x
15 1 2 3 4 5 6 7 8 mdetfval1 D = m B R p P Y S p · ˙ U x N p x m x
16 ovex R p P Y S p · ˙ U x N p x M x V
17 14 15 16 fvmpt M B D M = R p P Y S p · ˙ U x N p x M x