Step |
Hyp |
Ref |
Expression |
1 |
|
resmhm2.u |
⊢ 𝑈 = ( 𝑇 ↾s 𝑋 ) |
2 |
|
mhmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → 𝑆 ∈ Mnd ) |
3 |
2
|
adantl |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝑆 ∈ Mnd ) |
4 |
1
|
submmnd |
⊢ ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) → 𝑈 ∈ Mnd ) |
5 |
4
|
ad2antrr |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝑈 ∈ Mnd ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
8 |
6 7
|
mhmf |
⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
10 |
9
|
ffnd |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝐹 Fn ( Base ‘ 𝑆 ) ) |
11 |
|
simplr |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ran 𝐹 ⊆ 𝑋 ) |
12 |
|
df-f |
⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝑋 ↔ ( 𝐹 Fn ( Base ‘ 𝑆 ) ∧ ran 𝐹 ⊆ 𝑋 ) ) |
13 |
10 11 12
|
sylanbrc |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝑋 ) |
14 |
1
|
submbas |
⊢ ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) → 𝑋 = ( Base ‘ 𝑈 ) ) |
15 |
14
|
ad2antrr |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝑋 = ( Base ‘ 𝑈 ) ) |
16 |
15
|
feq3d |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝑋 ↔ 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ) ) |
17 |
13 16
|
mpbid |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ) |
18 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
19 |
|
eqid |
⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 ) |
20 |
6 18 19
|
mhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
21 |
20
|
3expb |
⊢ ( ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
22 |
21
|
adantll |
⊢ ( ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
23 |
1 19
|
ressplusg |
⊢ ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) → ( +g ‘ 𝑇 ) = ( +g ‘ 𝑈 ) ) |
24 |
23
|
ad3antrrr |
⊢ ( ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( +g ‘ 𝑇 ) = ( +g ‘ 𝑈 ) ) |
25 |
24
|
oveqd |
⊢ ( ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
26 |
22 25
|
eqtrd |
⊢ ( ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
27 |
26
|
ralrimivva |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
28 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
29 |
|
eqid |
⊢ ( 0g ‘ 𝑇 ) = ( 0g ‘ 𝑇 ) |
30 |
28 29
|
mhm0 |
⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) |
31 |
30
|
adantl |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) |
32 |
1 29
|
subm0 |
⊢ ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) → ( 0g ‘ 𝑇 ) = ( 0g ‘ 𝑈 ) ) |
33 |
32
|
ad2antrr |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ( 0g ‘ 𝑇 ) = ( 0g ‘ 𝑈 ) ) |
34 |
31 33
|
eqtrd |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑈 ) ) |
35 |
17 27 34
|
3jca |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑈 ) ) ) |
36 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
37 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
38 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
39 |
6 36 18 37 28 38
|
ismhm |
⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑈 ) ↔ ( ( 𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd ) ∧ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑈 ) ) ) ) |
40 |
3 5 35 39
|
syl21anbrc |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MndHom 𝑈 ) ) |
41 |
1
|
resmhm2 |
⊢ ( ( 𝐹 ∈ ( 𝑆 MndHom 𝑈 ) ∧ 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) |
42 |
41
|
ancoms |
⊢ ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑈 ) ) → 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) |
43 |
42
|
adantlr |
⊢ ( ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MndHom 𝑈 ) ) → 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ) |
44 |
40 43
|
impbida |
⊢ ( ( 𝑋 ∈ ( SubMnd ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) → ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ↔ 𝐹 ∈ ( 𝑆 MndHom 𝑈 ) ) ) |