Metamath Proof Explorer


Theorem mdetpmtr1

Description: The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (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
mdetpmtr1.e E = i N , j N P i M j
Assertion mdetpmtr1 R CRing N Fin M B P G D M = Z S P · ˙ 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 mdetpmtr1.e E = i N , j N P i M j
9 eqid Base R = Base R
10 eqid 0 R = 0 R
11 eqid + R = + R
12 crngring R CRing R Ring
13 12 ad2antrr R CRing N Fin M B P G R Ring
14 4 fvexi G V
15 14 a1i R CRing N Fin M B P G G V
16 simplr R CRing N Fin M B P G N Fin
17 5 4 psgndmfi N Fin S Fn G
18 fnfun S Fn G Fun S
19 16 17 18 3syl R CRing N Fin M B P G Fun S
20 simprr R CRing N Fin M B P G P G
21 fndm S Fn G dom S = G
22 16 17 21 3syl R CRing N Fin M B P G dom S = G
23 20 22 eleqtrrd R CRing N Fin M B P G P dom S
24 fvco Fun S P dom S Z S P = Z S P
25 19 23 24 syl2anc R CRing N Fin M B P G Z S P = Z S P
26 4 5 6 zrhpsgnelbas R Ring N Fin P G Z S P Base R
27 13 16 20 26 syl3anc R CRing N Fin M B P G Z S P Base R
28 25 27 eqeltrd R CRing N Fin M B P G Z S P Base R
29 13 adantr R CRing N Fin M B P G p G R Ring
30 4 5 cofipsgn N Fin p G Z S p = Z S p
31 16 30 sylan R CRing N Fin M B P G p G Z S p = Z S p
32 simpllr R CRing N Fin M B P G p G N Fin
33 simpr R CRing N Fin M B P G p G p G
34 4 5 6 zrhpsgnelbas R Ring N Fin p G Z S p Base R
35 29 32 33 34 syl3anc R CRing N Fin M B P G p G Z S p Base R
36 31 35 eqeltrd R CRing N Fin M B P G p G Z S p Base R
37 eqid mulGrp R = mulGrp R
38 37 9 mgpbas Base R = Base mulGrp R
39 37 crngmgp R CRing mulGrp R CMnd
40 39 ad3antrrr R CRing N Fin M B P G p G mulGrp R CMnd
41 eqid SymGrp N = SymGrp N
42 41 4 symgfv p G x N p x N
43 42 adantll R CRing N Fin M B P G p G x N p x N
44 simpr R CRing N Fin M B P G p G x N x N
45 simpll R CRing N Fin M B P G R CRing
46 simp1rr R CRing N Fin M B P G i N j N P G
47 simp2 R CRing N Fin M B P G i N j N i N
48 41 4 symgfv P G i N P i N
49 46 47 48 syl2anc R CRing N Fin M B P G i N j N P i N
50 simp3 R CRing N Fin M B P G i N j N j N
51 simp1rl R CRing N Fin M B P G i N j N M B
52 1 9 2 49 50 51 matecld R CRing N Fin M B P G i N j N P i M j Base R
53 1 9 2 16 45 52 matbas2d R CRing N Fin M B P G i N , j N P i M j B
54 8 53 eqeltrid R CRing N Fin M B P G E B
55 54 ad2antrr R CRing N Fin M B P G p G x N E B
56 1 9 2 43 44 55 matecld R CRing N Fin M B P G p G x N p x E x Base R
57 56 ralrimiva R CRing N Fin M B P G p G x N p x E x Base R
58 38 40 32 57 gsummptcl R CRing N Fin M B P G p G mulGrp R x N p x E x Base R
59 9 7 ringcl R Ring Z S p Base R mulGrp R x N p x E x Base R Z S p · ˙ mulGrp R x N p x E x Base R
60 29 36 58 59 syl3anc R CRing N Fin M B P G p G Z S p · ˙ mulGrp R x N p x E x Base R
61 eqid p G Z S p · ˙ mulGrp R x N p x E x = p G Z S p · ˙ mulGrp R x N p x E x
62 41 4 symgbasfi N Fin G Fin
63 16 62 syl R CRing N Fin M B P G G Fin
64 ovexd R CRing N Fin M B P G p G Z S p · ˙ mulGrp R x N p x E x V
65 fvexd R CRing N Fin M B P G 0 R V
66 61 63 64 65 fsuppmptdm R CRing N Fin M B P G finSupp 0 R p G Z S p · ˙ mulGrp R x N p x E x
67 9 10 11 7 13 15 28 60 66 gsummulc2 R CRing N Fin M B P G R p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = Z S P · ˙ R p G Z S p · ˙ mulGrp R x N p x E x
68 nfcv _ q Z S P p · ˙ mulGrp R x N P p x M x
69 fveq2 q = P p Z S q = Z S P p
70 fveq1 q = P p q x = P p x
71 70 oveq1d q = P p q x M x = P p x M x
72 71 mpteq2dv q = P p x N q x M x = x N P p x M x
73 72 oveq2d q = P p mulGrp R x N q x M x = mulGrp R x N P p x M x
74 69 73 oveq12d q = P p Z S q · ˙ mulGrp R x N q x M x = Z S P p · ˙ mulGrp R x N P p x M x
75 ringcmn R Ring R CMnd
76 13 75 syl R CRing N Fin M B P G R CMnd
77 ssidd R CRing N Fin M B P G Base R Base R
78 13 adantr R CRing N Fin M B P G q G R Ring
79 4 5 cofipsgn N Fin q G Z S q = Z S q
80 16 79 sylan R CRing N Fin M B P G q G Z S q = Z S q
81 simpllr R CRing N Fin M B P G q G N Fin
82 simpr R CRing N Fin M B P G q G q G
83 4 5 6 zrhpsgnelbas R Ring N Fin q G Z S q Base R
84 78 81 82 83 syl3anc R CRing N Fin M B P G q G Z S q Base R
85 80 84 eqeltrd R CRing N Fin M B P G q G Z S q Base R
86 39 ad3antrrr R CRing N Fin M B P G q G mulGrp R CMnd
87 41 4 symgfv q G x N q x N
88 87 adantll R CRing N Fin M B P G q G x N q x N
89 simpr R CRing N Fin M B P G q G x N x N
90 simprl R CRing N Fin M B P G M B
91 90 ad2antrr R CRing N Fin M B P G q G x N M B
92 1 9 2 88 89 91 matecld R CRing N Fin M B P G q G x N q x M x Base R
93 92 ralrimiva R CRing N Fin M B P G q G x N q x M x Base R
94 38 86 81 93 gsummptcl R CRing N Fin M B P G q G mulGrp R x N q x M x Base R
95 9 7 ringcl R Ring Z S q Base R mulGrp R x N q x M x Base R Z S q · ˙ mulGrp R x N q x M x Base R
96 78 85 94 95 syl3anc R CRing N Fin M B P G q G Z S q · ˙ mulGrp R x N q x M x Base R
97 eqid + SymGrp N = + SymGrp N
98 41 4 97 symgov P G p G P + SymGrp N p = P p
99 41 4 97 symgcl P G p G P + SymGrp N p G
100 98 99 eqeltrrd P G p G P p G
101 20 100 sylan R CRing N Fin M B P G p G P p G
102 20 adantr R CRing N Fin M B P G q G P G
103 4 symgfcoeu N Fin P G q G ∃! p G q = P p
104 81 102 82 103 syl3anc R CRing N Fin M B P G q G ∃! p G q = P p
105 68 9 10 74 76 63 77 96 101 104 gsummptf1o R CRing N Fin M B P G R q G Z S q · ˙ mulGrp R x N q x M x = R p G Z S P p · ˙ mulGrp R x N P p x M x
106 3 1 2 4 6 5 7 37 mdetleib M B D M = R q G Z S q · ˙ mulGrp R x N q x M x
107 106 ad2antrl R CRing N Fin M B P G D M = R q G Z S q · ˙ mulGrp R x N q x M x
108 28 adantr R CRing N Fin M B P G p G Z S P Base R
109 9 7 ringass R Ring Z S P Base R Z S p Base R mulGrp R x N p x E x Base R Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x
110 29 108 36 58 109 syl13anc R CRing N Fin M B P G p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x
111 25 adantr R CRing N Fin M B P G p G Z S P = Z S P
112 111 31 oveq12d R CRing N Fin M B P G p G Z S P · ˙ Z S p = Z S P · ˙ Z S p
113 4 5 cofipsgn N Fin P p G Z S P p = Z S P p
114 32 101 113 syl2anc R CRing N Fin M B P G p G Z S P p = Z S P p
115 20 adantr R CRing N Fin M B P G p G P G
116 41 5 4 psgnco N Fin P G p G S P p = S P S p
117 32 115 33 116 syl3anc R CRing N Fin M B P G p G S P p = S P S p
118 117 fveq2d R CRing N Fin M B P G p G Z S P p = Z S P S p
119 6 zrhrhm R Ring Z ring RingHom R
120 13 119 syl R CRing N Fin M B P G Z ring RingHom R
121 120 adantr R CRing N Fin M B P G p G Z ring RingHom R
122 1z 1
123 neg1z 1
124 prssi 1 1 1 1
125 122 123 124 mp2an 1 1
126 4 5 psgnran N Fin P G S P 1 1
127 32 115 126 syl2anc R CRing N Fin M B P G p G S P 1 1
128 125 127 sselid R CRing N Fin M B P G p G S P
129 4 5 psgnran N Fin p G S p 1 1
130 16 129 sylan R CRing N Fin M B P G p G S p 1 1
131 125 130 sselid R CRing N Fin M B P G p G S p
132 zringbas = Base ring
133 zringmulr × = ring
134 132 133 7 rhmmul Z ring RingHom R S P S p Z S P S p = Z S P · ˙ Z S p
135 121 128 131 134 syl3anc R CRing N Fin M B P G p G Z S P S p = Z S P · ˙ Z S p
136 114 118 135 3eqtrrd R CRing N Fin M B P G p G Z S P · ˙ Z S p = Z S P p
137 112 136 eqtrd R CRing N Fin M B P G p G Z S P · ˙ Z S p = Z S P p
138 8 a1i R CRing N Fin M B P G p G x N E = i N , j N P i M j
139 simprl R CRing N Fin M B P G p G x N i = p x j = x i = p x
140 139 fveq2d R CRing N Fin M B P G p G x N i = p x j = x P i = P p x
141 simpllr R CRing N Fin M B P G p G x N i = p x j = x p G
142 41 4 symgbasf p G p : N N
143 ffun p : N N Fun p
144 141 142 143 3syl R CRing N Fin M B P G p G x N i = p x j = x Fun p
145 simplr R CRing N Fin M B P G p G x N i = p x j = x x N
146 fdm p : N N dom p = N
147 141 142 146 3syl R CRing N Fin M B P G p G x N i = p x j = x dom p = N
148 145 147 eleqtrrd R CRing N Fin M B P G p G x N i = p x j = x x dom p
149 fvco Fun p x dom p P p x = P p x
150 144 148 149 syl2anc R CRing N Fin M B P G p G x N i = p x j = x P p x = P p x
151 140 150 eqtr4d R CRing N Fin M B P G p G x N i = p x j = x P i = P p x
152 simprr R CRing N Fin M B P G p G x N i = p x j = x j = x
153 151 152 oveq12d R CRing N Fin M B P G p G x N i = p x j = x P i M j = P p x M x
154 ovexd R CRing N Fin M B P G p G x N P p x M x V
155 138 153 43 44 154 ovmpod R CRing N Fin M B P G p G x N p x E x = P p x M x
156 155 mpteq2dva R CRing N Fin M B P G p G x N p x E x = x N P p x M x
157 156 oveq2d R CRing N Fin M B P G p G mulGrp R x N p x E x = mulGrp R x N P p x M x
158 137 157 oveq12d R CRing N Fin M B P G p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = Z S P p · ˙ mulGrp R x N P p x M x
159 110 158 eqtr3d R CRing N Fin M B P G p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = Z S P p · ˙ mulGrp R x N P p x M x
160 159 mpteq2dva R CRing N Fin M B P G p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = p G Z S P p · ˙ mulGrp R x N P p x M x
161 160 oveq2d R CRing N Fin M B P G R p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x = R p G Z S P p · ˙ mulGrp R x N P p x M x
162 105 107 161 3eqtr4d R CRing N Fin M B P G D M = R p G Z S P · ˙ Z S p · ˙ mulGrp R x N p x E x
163 3 1 2 4 6 5 7 37 mdetleib E B D E = R p G Z S p · ˙ mulGrp R x N p x E x
164 54 163 syl R CRing N Fin M B P G D E = R p G Z S p · ˙ mulGrp R x N p x E x
165 164 oveq2d R CRing N Fin M B P G Z S P · ˙ D E = Z S P · ˙ R p G Z S p · ˙ mulGrp R x N p x E x
166 67 162 165 3eqtr4d R CRing N Fin M B P G D M = Z S P · ˙ D E