Step |
Hyp |
Ref |
Expression |
1 |
|
resmgmhm2.u |
⊢ 𝑈 = ( 𝑇 ↾s 𝑋 ) |
2 |
|
mgmhmrcl |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ) |
3 |
2
|
simpld |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → 𝑆 ∈ Mgm ) |
4 |
3
|
adantl |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝑆 ∈ Mgm ) |
5 |
1
|
submgmmgm |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑈 ∈ Mgm ) |
6 |
5
|
ad2antrr |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝑈 ∈ Mgm ) |
7 |
4 6
|
jca |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → ( 𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
10 |
8 9
|
mgmhmf |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
12 |
11
|
ffnd |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝐹 Fn ( Base ‘ 𝑆 ) ) |
13 |
|
simplr |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → ran 𝐹 ⊆ 𝑋 ) |
14 |
|
df-f |
⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝑋 ↔ ( 𝐹 Fn ( Base ‘ 𝑆 ) ∧ ran 𝐹 ⊆ 𝑋 ) ) |
15 |
12 13 14
|
sylanbrc |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝑋 ) |
16 |
1
|
submgmbas |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑋 = ( Base ‘ 𝑈 ) ) |
17 |
16
|
ad2antrr |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝑋 = ( Base ‘ 𝑈 ) ) |
18 |
17
|
feq3d |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ 𝑋 ↔ 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ) ) |
19 |
15 18
|
mpbid |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ) |
20 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
21 |
|
eqid |
⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 ) |
22 |
8 20 21
|
mgmhmlin |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
23 |
22
|
3expb |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
24 |
23
|
adantll |
⊢ ( ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) |
25 |
1 21
|
ressplusg |
⊢ ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → ( +g ‘ 𝑇 ) = ( +g ‘ 𝑈 ) ) |
26 |
25
|
ad3antrrr |
⊢ ( ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( +g ‘ 𝑇 ) = ( +g ‘ 𝑈 ) ) |
27 |
26
|
oveqd |
⊢ ( ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
28 |
24 27
|
eqtrd |
⊢ ( ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
29 |
28
|
ralrimivva |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) |
30 |
19 29
|
jca |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
31 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
32 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
33 |
8 31 20 32
|
ismgmhm |
⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ↔ ( ( 𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
34 |
7 30 33
|
sylanbrc |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ) |
35 |
1
|
resmgmhm2 |
⊢ ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) |
36 |
35
|
ancoms |
⊢ ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) |
37 |
36
|
adantlr |
⊢ ( ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) ∧ 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ) |
38 |
34 37
|
impbida |
⊢ ( ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ∧ ran 𝐹 ⊆ 𝑋 ) → ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ↔ 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ) ) |