Step |
Hyp |
Ref |
Expression |
1 |
|
dvrcl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
dvrcl.o |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
3 |
|
dvrcl.d |
⊢ / = ( /r ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( invr ‘ 𝑅 ) = ( invr ‘ 𝑅 ) |
6 |
1 4 2 5 3
|
dvrval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( 𝑋 / 𝑌 ) = ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ) ) |
7 |
6
|
3adant1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( 𝑋 / 𝑌 ) = ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ) ) |
8 |
2 5 1
|
ringinvcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈 ) → ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ∈ 𝐵 ) |
9 |
8
|
3adant2 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ∈ 𝐵 ) |
10 |
1 4
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ) ∈ 𝐵 ) |
11 |
9 10
|
syld3an3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( 𝑋 ( .r ‘ 𝑅 ) ( ( invr ‘ 𝑅 ) ‘ 𝑌 ) ) ∈ 𝐵 ) |
12 |
7 11
|
eqeltrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( 𝑋 / 𝑌 ) ∈ 𝐵 ) |