Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsunit.1 |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
2 |
|
dvdsunit.3 |
⊢ ∥ = ( ∥r ‘ 𝑅 ) |
3 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
5 |
4 2
|
dvdsrtr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∥ 𝑋 ∧ 𝑋 ∥ ( 1r ‘ 𝑅 ) ) → 𝑌 ∥ ( 1r ‘ 𝑅 ) ) |
6 |
5
|
3expia |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∥ 𝑋 ) → ( 𝑋 ∥ ( 1r ‘ 𝑅 ) → 𝑌 ∥ ( 1r ‘ 𝑅 ) ) ) |
7 |
3 6
|
sylan |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ) → ( 𝑋 ∥ ( 1r ‘ 𝑅 ) → 𝑌 ∥ ( 1r ‘ 𝑅 ) ) ) |
8 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
9 |
1 8 2
|
crngunit |
⊢ ( 𝑅 ∈ CRing → ( 𝑋 ∈ 𝑈 ↔ 𝑋 ∥ ( 1r ‘ 𝑅 ) ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ) → ( 𝑋 ∈ 𝑈 ↔ 𝑋 ∥ ( 1r ‘ 𝑅 ) ) ) |
11 |
1 8 2
|
crngunit |
⊢ ( 𝑅 ∈ CRing → ( 𝑌 ∈ 𝑈 ↔ 𝑌 ∥ ( 1r ‘ 𝑅 ) ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ) → ( 𝑌 ∈ 𝑈 ↔ 𝑌 ∥ ( 1r ‘ 𝑅 ) ) ) |
13 |
7 10 12
|
3imtr4d |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ) → ( 𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈 ) ) |
14 |
13
|
3impia |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ∧ 𝑋 ∈ 𝑈 ) → 𝑌 ∈ 𝑈 ) |