Step |
Hyp |
Ref |
Expression |
1 |
|
resmgmhm2.u |
⊢ 𝑈 = ( 𝑇 ↾s 𝑋 ) |
2 |
|
mgmhmrcl |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) → ( 𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm ) ) |
3 |
2
|
simpld |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) → 𝑆 ∈ Mgm ) |
4 |
|
submgmrcl |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑇 ∈ Mgm ) |
5 |
3 4
|
anim12i |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
8 |
6 7
|
mgmhmf |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ) |
9 |
1
|
submgmbas |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑋 = ( Base ‘ 𝑈 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
11 |
10
|
submgmss |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑋 ⊆ ( Base ‘ 𝑇 ) ) |
12 |
9 11
|
eqsstrrd |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → ( Base ‘ 𝑈 ) ⊆ ( Base ‘ 𝑇 ) ) |
13 |
|
fss |
⊢ ( ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ∧ ( Base ‘ 𝑈 ) ⊆ ( Base ‘ 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
14 |
8 12 13
|
syl2an |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
15 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
16 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
17 |
6 15 16
|
mgmhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
18 |
17
|
3expb |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
19 |
18
|
adantlr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
20 |
|
eqid |
⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 ) |
21 |
1 20
|
ressplusg |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → ( +g ‘ 𝑇 ) = ( +g ‘ 𝑈 ) ) |
22 |
21
|
ad2antlr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( +g ‘ 𝑇 ) = ( +g ‘ 𝑈 ) ) |
23 |
22
|
oveqd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
24 |
19 23
|
eqtr4d |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
25 |
24
|
ralrimivva |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
26 |
14 25
|
jca |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
27 |
6 10 15 20
|
ismgmhm |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
28 |
5 26 27
|
sylanbrc |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) |