Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
2 |
1
|
idghm |
⊢ ( 𝑅 ∈ Grp → ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) ) |
3 |
|
cnvresid |
⊢ ◡ ( I ↾ ( Base ‘ 𝑅 ) ) = ( I ↾ ( Base ‘ 𝑅 ) ) |
4 |
3 2
|
eqeltrid |
⊢ ( 𝑅 ∈ Grp → ◡ ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) ) |
5 |
|
isgim2 |
⊢ ( ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpIso 𝑅 ) ↔ ( ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) ∧ ◡ ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) ) ) |
6 |
2 4 5
|
sylanbrc |
⊢ ( 𝑅 ∈ Grp → ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpIso 𝑅 ) ) |
7 |
|
brgici |
⊢ ( ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpIso 𝑅 ) → 𝑅 ≃𝑔 𝑅 ) |
8 |
6 7
|
syl |
⊢ ( 𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅 ) |