Step |
Hyp |
Ref |
Expression |
1 |
|
drngnzr |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ NzRing ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
5 |
2 3 4
|
isdrng |
⊢ ( 𝑅 ∈ DivRing ↔ ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) ) |
6 |
5
|
simprbi |
⊢ ( 𝑅 ∈ DivRing → ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) |
7 |
|
drngring |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring ) |
8 |
|
eqid |
⊢ ( RLReg ‘ 𝑅 ) = ( RLReg ‘ 𝑅 ) |
9 |
8 3
|
unitrrg |
⊢ ( 𝑅 ∈ Ring → ( Unit ‘ 𝑅 ) ⊆ ( RLReg ‘ 𝑅 ) ) |
10 |
7 9
|
syl |
⊢ ( 𝑅 ∈ DivRing → ( Unit ‘ 𝑅 ) ⊆ ( RLReg ‘ 𝑅 ) ) |
11 |
6 10
|
eqsstrrd |
⊢ ( 𝑅 ∈ DivRing → ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ⊆ ( RLReg ‘ 𝑅 ) ) |
12 |
2 8 4
|
isdomn2 |
⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ⊆ ( RLReg ‘ 𝑅 ) ) ) |
13 |
1 11 12
|
sylanbrc |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Domn ) |