Step |
Hyp |
Ref |
Expression |
0 |
|
cmdat |
⊢ maDet |
1 |
|
vn |
⊢ 𝑛 |
2 |
|
cvv |
⊢ V |
3 |
|
vr |
⊢ 𝑟 |
4 |
|
vm |
⊢ 𝑚 |
5 |
|
cbs |
⊢ Base |
6 |
1
|
cv |
⊢ 𝑛 |
7 |
|
cmat |
⊢ Mat |
8 |
3
|
cv |
⊢ 𝑟 |
9 |
6 8 7
|
co |
⊢ ( 𝑛 Mat 𝑟 ) |
10 |
9 5
|
cfv |
⊢ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) |
11 |
|
cgsu |
⊢ Σg |
12 |
|
vp |
⊢ 𝑝 |
13 |
|
csymg |
⊢ SymGrp |
14 |
6 13
|
cfv |
⊢ ( SymGrp ‘ 𝑛 ) |
15 |
14 5
|
cfv |
⊢ ( Base ‘ ( SymGrp ‘ 𝑛 ) ) |
16 |
|
czrh |
⊢ ℤRHom |
17 |
8 16
|
cfv |
⊢ ( ℤRHom ‘ 𝑟 ) |
18 |
|
cpsgn |
⊢ pmSgn |
19 |
6 18
|
cfv |
⊢ ( pmSgn ‘ 𝑛 ) |
20 |
17 19
|
ccom |
⊢ ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) |
21 |
12
|
cv |
⊢ 𝑝 |
22 |
21 20
|
cfv |
⊢ ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) |
23 |
|
cmulr |
⊢ .r |
24 |
8 23
|
cfv |
⊢ ( .r ‘ 𝑟 ) |
25 |
|
cmgp |
⊢ mulGrp |
26 |
8 25
|
cfv |
⊢ ( mulGrp ‘ 𝑟 ) |
27 |
|
vx |
⊢ 𝑥 |
28 |
27
|
cv |
⊢ 𝑥 |
29 |
28 21
|
cfv |
⊢ ( 𝑝 ‘ 𝑥 ) |
30 |
4
|
cv |
⊢ 𝑚 |
31 |
29 28 30
|
co |
⊢ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) |
32 |
27 6 31
|
cmpt |
⊢ ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) |
33 |
26 32 11
|
co |
⊢ ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) |
34 |
22 33 24
|
co |
⊢ ( ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) ( .r ‘ 𝑟 ) ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) ) |
35 |
12 15 34
|
cmpt |
⊢ ( 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑛 ) ) ↦ ( ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) ( .r ‘ 𝑟 ) ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) ) ) |
36 |
8 35 11
|
co |
⊢ ( 𝑟 Σg ( 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑛 ) ) ↦ ( ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) ( .r ‘ 𝑟 ) ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) ) ) ) |
37 |
4 10 36
|
cmpt |
⊢ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑟 Σg ( 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑛 ) ) ↦ ( ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) ( .r ‘ 𝑟 ) ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) ) ) ) ) |
38 |
1 3 2 2 37
|
cmpo |
⊢ ( 𝑛 ∈ V , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑟 Σg ( 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑛 ) ) ↦ ( ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) ( .r ‘ 𝑟 ) ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) ) ) ) ) ) |
39 |
0 38
|
wceq |
⊢ maDet = ( 𝑛 ∈ V , 𝑟 ∈ V ↦ ( 𝑚 ∈ ( Base ‘ ( 𝑛 Mat 𝑟 ) ) ↦ ( 𝑟 Σg ( 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑛 ) ) ↦ ( ( ( ( ℤRHom ‘ 𝑟 ) ∘ ( pmSgn ‘ 𝑛 ) ) ‘ 𝑝 ) ( .r ‘ 𝑟 ) ( ( mulGrp ‘ 𝑟 ) Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑝 ‘ 𝑥 ) 𝑚 𝑥 ) ) ) ) ) ) ) ) |