| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | id 22 | . 2
⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mnd) | 
| 2 |  | f1oi 6886 | . . . 4
⊢ ( I
↾ 𝐵):𝐵–1-1-onto→𝐵 | 
| 3 |  | f1of 6848 | . . . 4
⊢ (( I
↾ 𝐵):𝐵–1-1-onto→𝐵 → ( I ↾ 𝐵):𝐵⟶𝐵) | 
| 4 | 2, 3 | mp1i 13 | . . 3
⊢ (𝑀 ∈ Mnd → ( I ↾
𝐵):𝐵⟶𝐵) | 
| 5 |  | idmhm.b | . . . . . . . 8
⊢ 𝐵 = (Base‘𝑀) | 
| 6 |  | eqid 2737 | . . . . . . . 8
⊢
(+g‘𝑀) = (+g‘𝑀) | 
| 7 | 5, 6 | mndcl 18755 | . . . . . . 7
⊢ ((𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) | 
| 8 | 7 | 3expb 1121 | . . . . . 6
⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) | 
| 9 |  | fvresi 7193 | . . . . . 6
⊢ ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏)) | 
| 10 | 8, 9 | syl 17 | . . . . 5
⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = (𝑎(+g‘𝑀)𝑏)) | 
| 11 |  | fvresi 7193 | . . . . . . 7
⊢ (𝑎 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑎) = 𝑎) | 
| 12 |  | fvresi 7193 | . . . . . . 7
⊢ (𝑏 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑏) = 𝑏) | 
| 13 | 11, 12 | oveqan12d 7450 | . . . . . 6
⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏)) | 
| 14 | 13 | adantl 481 | . . . . 5
⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) = (𝑎(+g‘𝑀)𝑏)) | 
| 15 | 10, 14 | eqtr4d 2780 | . . . 4
⊢ ((𝑀 ∈ Mnd ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏))) | 
| 16 | 15 | ralrimivva 3202 | . . 3
⊢ (𝑀 ∈ Mnd → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏))) | 
| 17 |  | eqid 2737 | . . . . 5
⊢
(0g‘𝑀) = (0g‘𝑀) | 
| 18 | 5, 17 | mndidcl 18762 | . . . 4
⊢ (𝑀 ∈ Mnd →
(0g‘𝑀)
∈ 𝐵) | 
| 19 |  | fvresi 7193 | . . . 4
⊢
((0g‘𝑀) ∈ 𝐵 → (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀)) | 
| 20 | 18, 19 | syl 17 | . . 3
⊢ (𝑀 ∈ Mnd → (( I ↾
𝐵)‘(0g‘𝑀)) = (0g‘𝑀)) | 
| 21 | 4, 16, 20 | 3jca 1129 | . 2
⊢ (𝑀 ∈ Mnd → (( I ↾
𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) ∧ (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀))) | 
| 22 | 5, 5, 6, 6, 17, 17 | ismhm 18798 | . 2
⊢ (( I
↾ 𝐵) ∈ (𝑀 MndHom 𝑀) ↔ ((𝑀 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (( I ↾ 𝐵):𝐵⟶𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( I ↾ 𝐵)‘(𝑎(+g‘𝑀)𝑏)) = ((( I ↾ 𝐵)‘𝑎)(+g‘𝑀)(( I ↾ 𝐵)‘𝑏)) ∧ (( I ↾ 𝐵)‘(0g‘𝑀)) = (0g‘𝑀)))) | 
| 23 | 1, 1, 21, 22 | syl21anbrc 1345 | 1
⊢ (𝑀 ∈ Mnd → ( I ↾
𝐵) ∈ (𝑀 MndHom 𝑀)) |