Step |
Hyp |
Ref |
Expression |
1 |
|
imassrn |
⊢ ( 𝐹 “ 𝑋 ) ⊆ ran 𝐹 |
2 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝑁 ) = ( Base ‘ 𝑁 ) |
4 |
2 3
|
mhmf |
⊢ ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) → 𝐹 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑁 ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → 𝐹 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑁 ) ) |
6 |
5
|
frnd |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ran 𝐹 ⊆ ( Base ‘ 𝑁 ) ) |
7 |
1 6
|
sstrid |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝐹 “ 𝑋 ) ⊆ ( Base ‘ 𝑁 ) ) |
8 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝑁 ) = ( 0g ‘ 𝑁 ) |
10 |
8 9
|
mhm0 |
⊢ ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) → ( 𝐹 ‘ ( 0g ‘ 𝑀 ) ) = ( 0g ‘ 𝑁 ) ) |
11 |
10
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝐹 ‘ ( 0g ‘ 𝑀 ) ) = ( 0g ‘ 𝑁 ) ) |
12 |
5
|
ffnd |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → 𝐹 Fn ( Base ‘ 𝑀 ) ) |
13 |
2
|
submss |
⊢ ( 𝑋 ∈ ( SubMnd ‘ 𝑀 ) → 𝑋 ⊆ ( Base ‘ 𝑀 ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → 𝑋 ⊆ ( Base ‘ 𝑀 ) ) |
15 |
8
|
subm0cl |
⊢ ( 𝑋 ∈ ( SubMnd ‘ 𝑀 ) → ( 0g ‘ 𝑀 ) ∈ 𝑋 ) |
16 |
15
|
adantl |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 0g ‘ 𝑀 ) ∈ 𝑋 ) |
17 |
|
fnfvima |
⊢ ( ( 𝐹 Fn ( Base ‘ 𝑀 ) ∧ 𝑋 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝑋 ) → ( 𝐹 ‘ ( 0g ‘ 𝑀 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
18 |
12 14 16 17
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝐹 ‘ ( 0g ‘ 𝑀 ) ) ∈ ( 𝐹 “ 𝑋 ) ) |
19 |
11 18
|
eqeltrrd |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 0g ‘ 𝑁 ) ∈ ( 𝐹 “ 𝑋 ) ) |
20 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ) |
21 |
|
eqidd |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) ) |
22 |
|
eqidd |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( +g ‘ 𝑁 ) = ( +g ‘ 𝑁 ) ) |
23 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
24 |
23
|
submcl |
⊢ ( ( 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ∈ 𝑋 ) |
25 |
24
|
3adant1l |
⊢ ( ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑧 ( +g ‘ 𝑀 ) 𝑥 ) ∈ 𝑋 ) |
26 |
20 14 21 22 25
|
mhmimalem |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ∀ 𝑥 ∈ ( 𝐹 “ 𝑋 ) ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 ( +g ‘ 𝑁 ) 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) |
27 |
|
mhmrcl2 |
⊢ ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) → 𝑁 ∈ Mnd ) |
28 |
27
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → 𝑁 ∈ Mnd ) |
29 |
|
eqid |
⊢ ( +g ‘ 𝑁 ) = ( +g ‘ 𝑁 ) |
30 |
3 9 29
|
issubm |
⊢ ( 𝑁 ∈ Mnd → ( ( 𝐹 “ 𝑋 ) ∈ ( SubMnd ‘ 𝑁 ) ↔ ( ( 𝐹 “ 𝑋 ) ⊆ ( Base ‘ 𝑁 ) ∧ ( 0g ‘ 𝑁 ) ∈ ( 𝐹 “ 𝑋 ) ∧ ∀ 𝑥 ∈ ( 𝐹 “ 𝑋 ) ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 ( +g ‘ 𝑁 ) 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) ) ) |
31 |
28 30
|
syl |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( ( 𝐹 “ 𝑋 ) ∈ ( SubMnd ‘ 𝑁 ) ↔ ( ( 𝐹 “ 𝑋 ) ⊆ ( Base ‘ 𝑁 ) ∧ ( 0g ‘ 𝑁 ) ∈ ( 𝐹 “ 𝑋 ) ∧ ∀ 𝑥 ∈ ( 𝐹 “ 𝑋 ) ∀ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ( 𝑥 ( +g ‘ 𝑁 ) 𝑦 ) ∈ ( 𝐹 “ 𝑋 ) ) ) ) |
32 |
7 19 26 31
|
mpbir3and |
⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝐹 “ 𝑋 ) ∈ ( SubMnd ‘ 𝑁 ) ) |