Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsr.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
dvdsr.2 |
⊢ ∥ = ( ∥r ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
4 |
1 2 3
|
dvdsr |
⊢ ( 𝑋 ∥ 𝑌 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 ) ) |
5 |
1 3
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 ) |
6 |
5
|
3expa |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 ) |
7 |
6
|
an32s |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 ) |
8 |
|
eleq1 |
⊢ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 ↔ 𝑌 ∈ 𝐵 ) ) |
9 |
7 8
|
syl5ibcom |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 → 𝑌 ∈ 𝐵 ) ) |
10 |
9
|
rexlimdva |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 → 𝑌 ∈ 𝐵 ) ) |
11 |
10
|
impr |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
12 |
4 11
|
sylan2b |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∥ 𝑌 ) → 𝑌 ∈ 𝐵 ) |