Step | Hyp | Ref
| Expression |
1 | | mgmhmrcl 44869 |
. . . . . 6
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) |
2 | 1 | simpld 498 |
. . . . 5
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm) |
3 | 2 | adantl 485 |
. . . 4
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm) |
4 | | resmgmhm2.u |
. . . . . 6
⊢ 𝑈 = (𝑇 ↾s 𝑋) |
5 | 4 | submgmmgm 44883 |
. . . . 5
⊢ (𝑋 ∈ (SubMgm‘𝑇) → 𝑈 ∈ Mgm) |
6 | 5 | ad2antrr 726 |
. . . 4
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑈 ∈ Mgm) |
7 | 3, 6 | jca 515 |
. . 3
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm)) |
8 | | eqid 2738 |
. . . . . . . . 9
⊢
(Base‘𝑆) =
(Base‘𝑆) |
9 | | eqid 2738 |
. . . . . . . . 9
⊢
(Base‘𝑇) =
(Base‘𝑇) |
10 | 8, 9 | mgmhmf 44872 |
. . . . . . . 8
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
11 | 10 | adantl 485 |
. . . . . . 7
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
12 | 11 | ffnd 6505 |
. . . . . 6
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆)) |
13 | | simplr 769 |
. . . . . 6
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ran 𝐹 ⊆ 𝑋) |
14 | | df-f 6343 |
. . . . . 6
⊢ (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹 ⊆ 𝑋)) |
15 | 12, 13, 14 | sylanbrc 586 |
. . . . 5
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋) |
16 | 4 | submgmbas 44884 |
. . . . . . 7
⊢ (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈)) |
17 | 16 | ad2antrr 726 |
. . . . . 6
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈)) |
18 | 17 | feq3d 6491 |
. . . . 5
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋 ↔ 𝐹:(Base‘𝑆)⟶(Base‘𝑈))) |
19 | 15, 18 | mpbid 235 |
. . . 4
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈)) |
20 | | eqid 2738 |
. . . . . . . . 9
⊢
(+g‘𝑆) = (+g‘𝑆) |
21 | | eqid 2738 |
. . . . . . . . 9
⊢
(+g‘𝑇) = (+g‘𝑇) |
22 | 8, 20, 21 | mgmhmlin 44874 |
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
23 | 22 | 3expb 1121 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
24 | 23 | adantll 714 |
. . . . . 6
⊢ ((((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 | 4, 21 | ressplusg 16715 |
. . . . . . . 8
⊢ (𝑋 ∈ (SubMgm‘𝑇) →
(+g‘𝑇) =
(+g‘𝑈)) |
26 | 25 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈)) |
27 | 26 | oveqd 7187 |
. . . . . 6
⊢ ((((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 2773 |
. . . . 5
⊢ ((((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 3103 |
. . . 4
⊢ (((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 515 |
. . 3
⊢ (((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 2738 |
. . . 4
⊢
(Base‘𝑈) =
(Base‘𝑈) |
32 | | eqid 2738 |
. . . 4
⊢
(+g‘𝑈) = (+g‘𝑈) |
33 | 8, 31, 20, 32 | ismgmhm 44871 |
. . 3
⊢ (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 586 |
. 2
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑈)) |
35 | 4 | resmgmhm2 44887 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) |
36 | 35 | ancoms 462 |
. . 3
⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) |
37 | 36 | adantlr 715 |
. 2
⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) |
38 | 34, 37 | impbida 801 |
1
⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈))) |