Step |
Hyp |
Ref |
Expression |
1 |
|
opprirred.1 |
⊢ 𝑆 = ( oppr ‘ 𝑅 ) |
2 |
|
opprirred.2 |
⊢ 𝐼 = ( Irred ‘ 𝑅 ) |
3 |
|
ralcom |
⊢ ( ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ↔ ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
7 |
4 5 1 6
|
opprmul |
⊢ ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) = ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) |
8 |
7
|
neeq1i |
⊢ ( ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ≠ 𝑥 ↔ ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ) |
9 |
8
|
2ralbii |
⊢ ( ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ≠ 𝑥 ↔ ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ) |
10 |
3 9
|
bitr4i |
⊢ ( ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ↔ ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ≠ 𝑥 ) |
11 |
10
|
anbi2i |
⊢ ( ( 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∧ ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ) ↔ ( 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ≠ 𝑥 ) ) |
12 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
13 |
|
eqid |
⊢ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) = ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) |
14 |
4 12 2 13 5
|
isirred |
⊢ ( 𝑥 ∈ 𝐼 ↔ ( 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∧ ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑧 ( .r ‘ 𝑅 ) 𝑦 ) ≠ 𝑥 ) ) |
15 |
1 4
|
opprbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑆 ) |
16 |
12 1
|
opprunit |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑆 ) |
17 |
|
eqid |
⊢ ( Irred ‘ 𝑆 ) = ( Irred ‘ 𝑆 ) |
18 |
15 16 17 13 6
|
isirred |
⊢ ( 𝑥 ∈ ( Irred ‘ 𝑆 ) ↔ ( 𝑥 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∧ ∀ 𝑦 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ∀ 𝑧 ∈ ( ( Base ‘ 𝑅 ) ∖ ( Unit ‘ 𝑅 ) ) ( 𝑦 ( .r ‘ 𝑆 ) 𝑧 ) ≠ 𝑥 ) ) |
19 |
11 14 18
|
3bitr4i |
⊢ ( 𝑥 ∈ 𝐼 ↔ 𝑥 ∈ ( Irred ‘ 𝑆 ) ) |
20 |
19
|
eqriv |
⊢ 𝐼 = ( Irred ‘ 𝑆 ) |