Step |
Hyp |
Ref |
Expression |
1 |
|
irredn0.i |
⊢ 𝐼 = ( Irred ‘ 𝑅 ) |
2 |
|
irredrmul.u |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
3 |
|
irredrmul.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( .r ‘ ( oppr ‘ 𝑅 ) ) = ( .r ‘ ( oppr ‘ 𝑅 ) ) |
7 |
4 3 5 6
|
opprmul |
⊢ ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) = ( 𝑋 · 𝑌 ) |
8 |
5
|
opprring |
⊢ ( 𝑅 ∈ Ring → ( oppr ‘ 𝑅 ) ∈ Ring ) |
9 |
5 1
|
opprirred |
⊢ 𝐼 = ( Irred ‘ ( oppr ‘ 𝑅 ) ) |
10 |
2 5
|
opprunit |
⊢ 𝑈 = ( Unit ‘ ( oppr ‘ 𝑅 ) ) |
11 |
9 10 6
|
irredrmul |
⊢ ( ( ( oppr ‘ 𝑅 ) ∈ Ring ∧ 𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) ∈ 𝐼 ) |
12 |
8 11
|
syl3an1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈 ) → ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) ∈ 𝐼 ) |
13 |
12
|
3com23 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼 ) → ( 𝑌 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑋 ) ∈ 𝐼 ) |
14 |
7 13
|
eqeltrrid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼 ) → ( 𝑋 · 𝑌 ) ∈ 𝐼 ) |