Step |
Hyp |
Ref |
Expression |
1 |
|
istrg.1 |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
2 |
|
elin |
⊢ ( 𝑅 ∈ ( TopGrp ∩ Ring ) ↔ ( 𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ) ) |
3 |
2
|
anbi1i |
⊢ ( ( 𝑅 ∈ ( TopGrp ∩ Ring ) ∧ 𝑀 ∈ TopMnd ) ↔ ( ( 𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ TopMnd ) ) |
4 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( mulGrp ‘ 𝑟 ) = ( mulGrp ‘ 𝑅 ) ) |
5 |
4 1
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( mulGrp ‘ 𝑟 ) = 𝑀 ) |
6 |
5
|
eleq1d |
⊢ ( 𝑟 = 𝑅 → ( ( mulGrp ‘ 𝑟 ) ∈ TopMnd ↔ 𝑀 ∈ TopMnd ) ) |
7 |
|
df-trg |
⊢ TopRing = { 𝑟 ∈ ( TopGrp ∩ Ring ) ∣ ( mulGrp ‘ 𝑟 ) ∈ TopMnd } |
8 |
6 7
|
elrab2 |
⊢ ( 𝑅 ∈ TopRing ↔ ( 𝑅 ∈ ( TopGrp ∩ Ring ) ∧ 𝑀 ∈ TopMnd ) ) |
9 |
|
df-3an |
⊢ ( ( 𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd ) ↔ ( ( 𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ) ∧ 𝑀 ∈ TopMnd ) ) |
10 |
3 8 9
|
3bitr4i |
⊢ ( 𝑅 ∈ TopRing ↔ ( 𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd ) ) |