Step | Hyp | Ref
| Expression |
1 | | lmhmlmod1 20098 |
. . 3
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) |
2 | | reslmhm.r |
. . . 4
⊢ 𝑅 = (𝑆 ↾s 𝑋) |
3 | | reslmhm.u |
. . . 4
⊢ 𝑈 = (LSubSp‘𝑆) |
4 | 2, 3 | lsslmod 20025 |
. . 3
⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod) |
5 | 1, 4 | sylan 583 |
. 2
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ LMod) |
6 | | lmhmlmod2 20097 |
. . 3
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod) |
7 | 6 | adantr 484 |
. 2
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑇 ∈ LMod) |
8 | | lmghm 20096 |
. . . 4
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
9 | 3 | lsssubg 20022 |
. . . . 5
⊢ ((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆)) |
10 | 1, 9 | sylan 583 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ (SubGrp‘𝑆)) |
11 | 2 | resghm 18666 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ (SubGrp‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇)) |
12 | 8, 10, 11 | syl2an2r 685 |
. . 3
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇)) |
13 | | eqid 2738 |
. . . . 5
⊢
(Scalar‘𝑆) =
(Scalar‘𝑆) |
14 | | eqid 2738 |
. . . . 5
⊢
(Scalar‘𝑇) =
(Scalar‘𝑇) |
15 | 13, 14 | lmhmsca 20095 |
. . . 4
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆)) |
16 | 2, 13 | resssca 16904 |
. . . 4
⊢ (𝑋 ∈ 𝑈 → (Scalar‘𝑆) = (Scalar‘𝑅)) |
17 | 15, 16 | sylan9eq 2799 |
. . 3
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑇) = (Scalar‘𝑅)) |
18 | | simpll 767 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
19 | | simprl 771 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘(Scalar‘𝑆))) |
20 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(Base‘𝑆) =
(Base‘𝑆) |
21 | 20, 3 | lssss 20001 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝑈 → 𝑋 ⊆ (Base‘𝑆)) |
22 | 21 | adantl 485 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 ⊆ (Base‘𝑆)) |
23 | 22 | adantr 484 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ⊆ (Base‘𝑆)) |
24 | 2, 20 | ressbas2 16819 |
. . . . . . . . . . . 12
⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑅)) |
25 | 22, 24 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑋 = (Base‘𝑅)) |
26 | 25 | eleq2d 2824 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑏 ∈ 𝑋 ↔ 𝑏 ∈ (Base‘𝑅))) |
27 | 26 | biimpar 481 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ 𝑋) |
28 | 27 | adantrl 716 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ 𝑋) |
29 | 23, 28 | sseldd 3917 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑆)) |
30 | | eqid 2738 |
. . . . . . . 8
⊢
(Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆)) |
31 | | eqid 2738 |
. . . . . . . 8
⊢ (
·𝑠 ‘𝑆) = ( ·𝑠
‘𝑆) |
32 | | eqid 2738 |
. . . . . . . 8
⊢ (
·𝑠 ‘𝑇) = ( ·𝑠
‘𝑇) |
33 | 13, 30, 20, 31, 32 | lmhmlin 20100 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)(𝐹‘𝑏))) |
34 | 18, 19, 29, 33 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)(𝐹‘𝑏))) |
35 | 1 | adantr 484 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → 𝑆 ∈ LMod) |
36 | 35 | adantr 484 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ LMod) |
37 | | simplr 769 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑋 ∈ 𝑈) |
38 | 13, 31, 30, 3 | lssvscl 20020 |
. . . . . . . 8
⊢ (((𝑆 ∈ LMod ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ 𝑋)) → (𝑎( ·𝑠
‘𝑆)𝑏) ∈ 𝑋) |
39 | 36, 37, 19, 28, 38 | syl22anc 839 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠
‘𝑆)𝑏) ∈ 𝑋) |
40 | 39 | fvresd 6756 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝐹‘(𝑎( ·𝑠
‘𝑆)𝑏))) |
41 | | fvres 6755 |
. . . . . . . 8
⊢ (𝑏 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑏) = (𝐹‘𝑏)) |
42 | 41 | oveq2d 7248 |
. . . . . . 7
⊢ (𝑏 ∈ 𝑋 → (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = (𝑎( ·𝑠
‘𝑇)(𝐹‘𝑏))) |
43 | 28, 42 | syl 17 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) = (𝑎( ·𝑠
‘𝑇)(𝐹‘𝑏))) |
44 | 34, 40, 43 | 3eqtr4d 2788 |
. . . . 5
⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑅))) → ((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))) |
45 | 44 | ralrimivva 3113 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))) |
46 | 16 | adantl 485 |
. . . . . 6
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Scalar‘𝑆) = (Scalar‘𝑅)) |
47 | 46 | fveq2d 6740 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (Base‘(Scalar‘𝑆)) =
(Base‘(Scalar‘𝑅))) |
48 | 2, 31 | ressvsca 16905 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝑈 → (
·𝑠 ‘𝑆) = ( ·𝑠
‘𝑅)) |
49 | 48 | adantl 485 |
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (
·𝑠 ‘𝑆) = ( ·𝑠
‘𝑅)) |
50 | 49 | oveqd 7249 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝑎( ·𝑠
‘𝑆)𝑏) = (𝑎( ·𝑠
‘𝑅)𝑏)) |
51 | 50 | fveqeq2d 6744 |
. . . . . 6
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) ↔ ((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑅)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))) |
52 | 51 | ralbidv 3119 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) ↔ ∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑅)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))) |
53 | 47, 52 | raleqbidv 3326 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑆)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)) ↔ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑅)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))) |
54 | 45, 53 | mpbid 235 |
. . 3
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑅)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))) |
55 | 12, 17, 54 | 3jca 1130 |
. 2
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → ((𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑅) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑅)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏)))) |
56 | | eqid 2738 |
. . 3
⊢
(Scalar‘𝑅) =
(Scalar‘𝑅) |
57 | | eqid 2738 |
. . 3
⊢
(Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅)) |
58 | | eqid 2738 |
. . 3
⊢
(Base‘𝑅) =
(Base‘𝑅) |
59 | | eqid 2738 |
. . 3
⊢ (
·𝑠 ‘𝑅) = ( ·𝑠
‘𝑅) |
60 | 56, 14, 57, 58, 59, 32 | islmhm 20092 |
. 2
⊢ ((𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇) ↔ ((𝑅 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ ((𝐹 ↾ 𝑋) ∈ (𝑅 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑅) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑅))∀𝑏 ∈ (Base‘𝑅)((𝐹 ↾ 𝑋)‘(𝑎( ·𝑠
‘𝑅)𝑏)) = (𝑎( ·𝑠
‘𝑇)((𝐹 ↾ 𝑋)‘𝑏))))) |
61 | 5, 7, 55, 60 | syl21anbrc 1346 |
1
⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇)) |