Step |
Hyp |
Ref |
Expression |
1 |
|
chrcl.c |
⊢ 𝐶 = ( chr ‘ 𝑅 ) |
2 |
|
chrid.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑅 ) |
3 |
|
chrid.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
1
|
chrcl |
⊢ ( 𝑅 ∈ Ring → 𝐶 ∈ ℕ0 ) |
5 |
4
|
nn0zd |
⊢ ( 𝑅 ∈ Ring → 𝐶 ∈ ℤ ) |
6 |
|
eqid |
⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 ) |
7 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
8 |
2 6 7
|
zrhmulg |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐶 ∈ ℤ ) → ( 𝐿 ‘ 𝐶 ) = ( 𝐶 ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) ) |
9 |
5 8
|
mpdan |
⊢ ( 𝑅 ∈ Ring → ( 𝐿 ‘ 𝐶 ) = ( 𝐶 ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) ) |
10 |
|
eqid |
⊢ ( od ‘ 𝑅 ) = ( od ‘ 𝑅 ) |
11 |
10 7 1
|
chrval |
⊢ ( ( od ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) = 𝐶 |
12 |
11
|
oveq1i |
⊢ ( ( ( od ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = ( 𝐶 ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
14 |
13 7
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
15 |
13 10 6 3
|
odid |
⊢ ( ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) → ( ( ( od ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 0 ) |
16 |
14 15
|
syl |
⊢ ( 𝑅 ∈ Ring → ( ( ( od ‘ 𝑅 ) ‘ ( 1r ‘ 𝑅 ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 0 ) |
17 |
12 16
|
syl5eqr |
⊢ ( 𝑅 ∈ Ring → ( 𝐶 ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) = 0 ) |
18 |
9 17
|
eqtrd |
⊢ ( 𝑅 ∈ Ring → ( 𝐿 ‘ 𝐶 ) = 0 ) |