Step |
Hyp |
Ref |
Expression |
1 |
|
isridl.u |
⊢ 𝑈 = ( LIdeal ‘ ( oppr ‘ 𝑅 ) ) |
2 |
|
isridl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
isridl.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 ) |
5 |
4
|
opprring |
⊢ ( 𝑅 ∈ Ring → ( oppr ‘ 𝑅 ) ∈ Ring ) |
6 |
4 2
|
opprbas |
⊢ 𝐵 = ( Base ‘ ( oppr ‘ 𝑅 ) ) |
7 |
|
eqid |
⊢ ( .r ‘ ( oppr ‘ 𝑅 ) ) = ( .r ‘ ( oppr ‘ 𝑅 ) ) |
8 |
1 6 7
|
dflidl2 |
⊢ ( ( oppr ‘ 𝑅 ) ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ) ) ) |
9 |
5 8
|
syl |
⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ) ) ) |
10 |
4
|
opprsubg |
⊢ ( SubGrp ‘ 𝑅 ) = ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) |
11 |
10
|
eqcomi |
⊢ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) = ( SubGrp ‘ 𝑅 ) |
12 |
11
|
a1i |
⊢ ( 𝑅 ∈ Ring → ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) = ( SubGrp ‘ 𝑅 ) ) |
13 |
12
|
eleq2d |
⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ↔ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) ) |
14 |
2 3 4 7
|
opprmul |
⊢ ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) = ( 𝑦 · 𝑥 ) |
15 |
14
|
eleq1i |
⊢ ( ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ( 𝑦 · 𝑥 ) ∈ 𝐼 ) |
16 |
15
|
a1i |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐼 ) → ( ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) |
17 |
16
|
ralbidva |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) |
18 |
17
|
ralbidva |
⊢ ( 𝑅 ∈ Ring → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) |
19 |
13 18
|
anbi12d |
⊢ ( 𝑅 ∈ Ring → ( ( 𝐼 ∈ ( SubGrp ‘ ( oppr ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ∈ 𝐼 ) ↔ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) ) |
20 |
9 19
|
bitrd |
⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) ) |