Step |
Hyp |
Ref |
Expression |
1 |
|
mdet1.d |
⊢ 𝐷 = ( 𝑁 maDet 𝑅 ) |
2 |
|
mdet1.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
mdet1.n |
⊢ 𝐼 = ( 1r ‘ 𝐴 ) |
4 |
|
mdet1.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
5 |
|
id |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ) |
6 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
7 |
6
|
anim1ci |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ) |
8 |
2
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
10 |
9 3
|
ringidcl |
⊢ ( 𝐴 ∈ Ring → 𝐼 ∈ ( Base ‘ 𝐴 ) ) |
11 |
7 8 10
|
3syl |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → 𝐼 ∈ ( Base ‘ 𝐴 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
13 |
12 4
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) ) |
14 |
6 13
|
syl |
⊢ ( 𝑅 ∈ CRing → 1 ∈ ( Base ‘ 𝑅 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → 1 ∈ ( Base ‘ 𝑅 ) ) |
16 |
5 11 15
|
jca32 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝐼 ∈ ( Base ‘ 𝐴 ) ∧ 1 ∈ ( Base ‘ 𝑅 ) ) ) ) |
17 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
18 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑁 ∈ Fin ) |
19 |
6
|
adantr |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → 𝑅 ∈ Ring ) |
20 |
19
|
adantr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑅 ∈ Ring ) |
21 |
|
simprl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑖 ∈ 𝑁 ) |
22 |
|
simprr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑗 ∈ 𝑁 ) |
23 |
2 4 17 18 20 21 22 3
|
mat1ov |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 𝐼 𝑗 ) = if ( 𝑖 = 𝑗 , 1 , ( 0g ‘ 𝑅 ) ) ) |
24 |
23
|
ralrimivva |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝐼 𝑗 ) = if ( 𝑖 = 𝑗 , 1 , ( 0g ‘ 𝑅 ) ) ) |
25 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
26 |
|
eqid |
⊢ ( .g ‘ ( mulGrp ‘ 𝑅 ) ) = ( .g ‘ ( mulGrp ‘ 𝑅 ) ) |
27 |
1 2 9 25 17 12 26
|
mdetdiagid |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) ∧ ( 𝐼 ∈ ( Base ‘ 𝐴 ) ∧ 1 ∈ ( Base ‘ 𝑅 ) ) ) → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝐼 𝑗 ) = if ( 𝑖 = 𝑗 , 1 , ( 0g ‘ 𝑅 ) ) → ( 𝐷 ‘ 𝐼 ) = ( ( ♯ ‘ 𝑁 ) ( .g ‘ ( mulGrp ‘ 𝑅 ) ) 1 ) ) ) |
28 |
16 24 27
|
sylc |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( 𝐷 ‘ 𝐼 ) = ( ( ♯ ‘ 𝑁 ) ( .g ‘ ( mulGrp ‘ 𝑅 ) ) 1 ) ) |
29 |
|
ringsrg |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ SRing ) |
30 |
6 29
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ SRing ) |
31 |
|
hashcl |
⊢ ( 𝑁 ∈ Fin → ( ♯ ‘ 𝑁 ) ∈ ℕ0 ) |
32 |
25 26 4
|
srg1expzeq1 |
⊢ ( ( 𝑅 ∈ SRing ∧ ( ♯ ‘ 𝑁 ) ∈ ℕ0 ) → ( ( ♯ ‘ 𝑁 ) ( .g ‘ ( mulGrp ‘ 𝑅 ) ) 1 ) = 1 ) |
33 |
30 31 32
|
syl2an |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( ( ♯ ‘ 𝑁 ) ( .g ‘ ( mulGrp ‘ 𝑅 ) ) 1 ) = 1 ) |
34 |
28 33
|
eqtrd |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ) → ( 𝐷 ‘ 𝐼 ) = 1 ) |