Step |
Hyp |
Ref |
Expression |
1 |
|
fnmgp |
⊢ mulGrp Fn V |
2 |
|
ssv |
⊢ Ring ⊆ V |
3 |
|
fnssres |
⊢ ( ( mulGrp Fn V ∧ Ring ⊆ V ) → ( mulGrp ↾ Ring ) Fn Ring ) |
4 |
1 2 3
|
mp2an |
⊢ ( mulGrp ↾ Ring ) Fn Ring |
5 |
|
fvres |
⊢ ( 𝑎 ∈ Ring → ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) = ( mulGrp ‘ 𝑎 ) ) |
6 |
|
eqid |
⊢ ( mulGrp ‘ 𝑎 ) = ( mulGrp ‘ 𝑎 ) |
7 |
6
|
ringmgp |
⊢ ( 𝑎 ∈ Ring → ( mulGrp ‘ 𝑎 ) ∈ Mnd ) |
8 |
5 7
|
eqeltrd |
⊢ ( 𝑎 ∈ Ring → ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) ∈ Mnd ) |
9 |
8
|
rgen |
⊢ ∀ 𝑎 ∈ Ring ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) ∈ Mnd |
10 |
|
ffnfv |
⊢ ( ( mulGrp ↾ Ring ) : Ring ⟶ Mnd ↔ ( ( mulGrp ↾ Ring ) Fn Ring ∧ ∀ 𝑎 ∈ Ring ( ( mulGrp ↾ Ring ) ‘ 𝑎 ) ∈ Mnd ) ) |
11 |
4 9 10
|
mpbir2an |
⊢ ( mulGrp ↾ Ring ) : Ring ⟶ Mnd |