Step |
Hyp |
Ref |
Expression |
1 |
|
ielefmnd.g |
⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
5 |
1
|
ielefmnd |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) |
6 |
1 2 4
|
efmndov |
⊢ ( ( ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ( +g ‘ 𝐺 ) 𝑓 ) = ( ( I ↾ 𝐴 ) ∘ 𝑓 ) ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ( +g ‘ 𝐺 ) 𝑓 ) = ( ( I ↾ 𝐴 ) ∘ 𝑓 ) ) |
8 |
1 2
|
efmndbasf |
⊢ ( 𝑓 ∈ ( Base ‘ 𝐺 ) → 𝑓 : 𝐴 ⟶ 𝐴 ) |
9 |
8
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → 𝑓 : 𝐴 ⟶ 𝐴 ) |
10 |
|
fcoi2 |
⊢ ( 𝑓 : 𝐴 ⟶ 𝐴 → ( ( I ↾ 𝐴 ) ∘ 𝑓 ) = 𝑓 ) |
11 |
9 10
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ∘ 𝑓 ) = 𝑓 ) |
12 |
7 11
|
eqtrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( ( I ↾ 𝐴 ) ( +g ‘ 𝐺 ) 𝑓 ) = 𝑓 ) |
13 |
5
|
anim1ci |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ∈ ( Base ‘ 𝐺 ) ∧ ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) ) |
14 |
1 2 4
|
efmndov |
⊢ ( ( 𝑓 ∈ ( Base ‘ 𝐺 ) ∧ ( I ↾ 𝐴 ) ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ( +g ‘ 𝐺 ) ( I ↾ 𝐴 ) ) = ( 𝑓 ∘ ( I ↾ 𝐴 ) ) ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ( +g ‘ 𝐺 ) ( I ↾ 𝐴 ) ) = ( 𝑓 ∘ ( I ↾ 𝐴 ) ) ) |
16 |
|
fcoi1 |
⊢ ( 𝑓 : 𝐴 ⟶ 𝐴 → ( 𝑓 ∘ ( I ↾ 𝐴 ) ) = 𝑓 ) |
17 |
9 16
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ∘ ( I ↾ 𝐴 ) ) = 𝑓 ) |
18 |
15 17
|
eqtrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑓 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ( +g ‘ 𝐺 ) ( I ↾ 𝐴 ) ) = 𝑓 ) |
19 |
2 3 4 5 12 18
|
ismgmid2 |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) = ( 0g ‘ 𝐺 ) ) |