Metamath Proof Explorer


Theorem mdetpmtr12

Description: The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020)

Ref Expression
Hypotheses mdetpmtr.a A = N Mat R
mdetpmtr.b B = Base A
mdetpmtr.d D = N maDet R
mdetpmtr.g G = Base SymGrp N
mdetpmtr.s S = pmSgn N
mdetpmtr.z Z = ℤRHom R
mdetpmtr.t · ˙ = R
mdetpmtr12.e E = i N , j N P i M Q j
mdetmptr12.r φ R CRing
mdetmptr12.n φ N Fin
mdetmptr12.m φ M B
mdetmptr12.p φ P G
mdetmptr12.q φ Q G
Assertion mdetpmtr12 φ D M = Z S P S Q · ˙ D E

Proof

Step Hyp Ref Expression
1 mdetpmtr.a A = N Mat R
2 mdetpmtr.b B = Base A
3 mdetpmtr.d D = N maDet R
4 mdetpmtr.g G = Base SymGrp N
5 mdetpmtr.s S = pmSgn N
6 mdetpmtr.z Z = ℤRHom R
7 mdetpmtr.t · ˙ = R
8 mdetpmtr12.e E = i N , j N P i M Q j
9 mdetmptr12.r φ R CRing
10 mdetmptr12.n φ N Fin
11 mdetmptr12.m φ M B
12 mdetmptr12.p φ P G
13 mdetmptr12.q φ Q G
14 fveq2 k = i P k = P i
15 14 oveq1d k = i P k M l = P i M l
16 oveq2 l = j P i M l = P i M j
17 15 16 cbvmpov k N , l N P k M l = i N , j N P i M j
18 1 2 3 4 5 6 7 17 mdetpmtr1 R CRing N Fin M B P G D M = Z S P · ˙ D k N , l N P k M l
19 9 10 11 12 18 syl22anc φ D M = Z S P · ˙ D k N , l N P k M l
20 eqid Base R = Base R
21 12 3ad2ant1 φ k N l N P G
22 simp2 φ k N l N k N
23 eqid SymGrp N = SymGrp N
24 23 4 symgfv P G k N P k N
25 21 22 24 syl2anc φ k N l N P k N
26 simp3 φ k N l N l N
27 11 3ad2ant1 φ k N l N M B
28 1 20 2 25 26 27 matecld φ k N l N P k M l Base R
29 1 20 2 10 9 28 matbas2d φ k N , l N P k M l B
30 eqid i N , j N i k N , l N P k M l Q j = i N , j N i k N , l N P k M l Q j
31 1 2 3 4 5 6 7 30 mdetpmtr2 R CRing N Fin k N , l N P k M l B Q G D k N , l N P k M l = Z S Q · ˙ D i N , j N i k N , l N P k M l Q j
32 9 10 29 13 31 syl22anc φ D k N , l N P k M l = Z S Q · ˙ D i N , j N i k N , l N P k M l Q j
33 simp2 φ i N j N i N
34 13 3ad2ant1 φ i N j N Q G
35 simp3 φ i N j N j N
36 23 4 symgfv Q G j N Q j N
37 34 35 36 syl2anc φ i N j N Q j N
38 oveq2 l = Q j P i M l = P i M Q j
39 eqid k N , l N P k M l = k N , l N P k M l
40 ovex P i M Q j V
41 15 38 39 40 ovmpo i N Q j N i k N , l N P k M l Q j = P i M Q j
42 33 37 41 syl2anc φ i N j N i k N , l N P k M l Q j = P i M Q j
43 42 mpoeq3dva φ i N , j N i k N , l N P k M l Q j = i N , j N P i M Q j
44 8 43 eqtr4id φ E = i N , j N i k N , l N P k M l Q j
45 44 fveq2d φ D E = D i N , j N i k N , l N P k M l Q j
46 45 oveq2d φ Z S Q · ˙ D E = Z S Q · ˙ D i N , j N i k N , l N P k M l Q j
47 32 46 eqtr4d φ D k N , l N P k M l = Z S Q · ˙ D E
48 47 oveq2d φ Z S P · ˙ D k N , l N P k M l = Z S P · ˙ Z S Q · ˙ D E
49 crngring R CRing R Ring
50 9 49 syl φ R Ring
51 4 5 6 zrhcopsgnelbas R Ring N Fin P G Z S P Base R
52 50 10 12 51 syl3anc φ Z S P Base R
53 4 5 6 zrhcopsgnelbas R Ring N Fin Q G Z S Q Base R
54 50 10 13 53 syl3anc φ Z S Q Base R
55 12 3ad2ant1 φ i N j N P G
56 23 4 symgfv P G i N P i N
57 55 33 56 syl2anc φ i N j N P i N
58 11 3ad2ant1 φ i N j N M B
59 1 20 2 57 37 58 matecld φ i N j N P i M Q j Base R
60 1 20 2 10 9 59 matbas2d φ i N , j N P i M Q j B
61 8 60 eqeltrid φ E B
62 3 1 2 20 mdetcl R CRing E B D E Base R
63 9 61 62 syl2anc φ D E Base R
64 20 7 ringass R Ring Z S P Base R Z S Q Base R D E Base R Z S P · ˙ Z S Q · ˙ D E = Z S P · ˙ Z S Q · ˙ D E
65 50 52 54 63 64 syl13anc φ Z S P · ˙ Z S Q · ˙ D E = Z S P · ˙ Z S Q · ˙ D E
66 4 5 cofipsgn N Fin P G Z S P = Z S P
67 10 12 66 syl2anc φ Z S P = Z S P
68 4 5 cofipsgn N Fin Q G Z S Q = Z S Q
69 10 13 68 syl2anc φ Z S Q = Z S Q
70 67 69 oveq12d φ Z S P · ˙ Z S Q = Z S P · ˙ Z S Q
71 6 zrhrhm R Ring Z ring RingHom R
72 50 71 syl φ Z ring RingHom R
73 1z 1
74 neg1z 1
75 prssi 1 1 1 1
76 73 74 75 mp2an 1 1
77 4 5 psgnran N Fin P G S P 1 1
78 10 12 77 syl2anc φ S P 1 1
79 76 78 sselid φ S P
80 4 5 psgnran N Fin Q G S Q 1 1
81 10 13 80 syl2anc φ S Q 1 1
82 76 81 sselid φ S Q
83 zringbas = Base ring
84 zringmulr × = ring
85 83 84 7 rhmmul Z ring RingHom R S P S Q Z S P S Q = Z S P · ˙ Z S Q
86 72 79 82 85 syl3anc φ Z S P S Q = Z S P · ˙ Z S Q
87 70 86 eqtr4d φ Z S P · ˙ Z S Q = Z S P S Q
88 87 oveq1d φ Z S P · ˙ Z S Q · ˙ D E = Z S P S Q · ˙ D E
89 65 88 eqtr3d φ Z S P · ˙ Z S Q · ˙ D E = Z S P S Q · ˙ D E
90 19 48 89 3eqtrd φ D M = Z S P S Q · ˙ D E