Step | Hyp | Ref
| Expression |
1 | | df-lmhm 18843 |
. . 3
⊢ LMHom =
(𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)))}) |
2 | 1 | elmpt2cl 6774 |
. 2
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod)) |
3 | | oveq12 6558 |
. . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇)) |
4 | | fvex 6113 |
. . . . . . . 8
⊢
(Scalar‘𝑠)
∈ V |
5 | 4 | a1i 11 |
. . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Scalar‘𝑠) ∈ V) |
6 | | simplr 788 |
. . . . . . . . . . 11
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑡 = 𝑇) |
7 | 6 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = (Scalar‘𝑇)) |
8 | | islmhm.l |
. . . . . . . . . 10
⊢ 𝐿 = (Scalar‘𝑇) |
9 | 7, 8 | syl6eqr 2662 |
. . . . . . . . 9
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = 𝐿) |
10 | | simpr 476 |
. . . . . . . . . . 11
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑠)) |
11 | | simpll 786 |
. . . . . . . . . . . 12
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑠 = 𝑆) |
12 | 11 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑠) = (Scalar‘𝑆)) |
13 | 10, 12 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑆)) |
14 | | islmhm.k |
. . . . . . . . . 10
⊢ 𝐾 = (Scalar‘𝑆) |
15 | 13, 14 | syl6eqr 2662 |
. . . . . . . . 9
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = 𝐾) |
16 | 9, 15 | eqeq12d 2625 |
. . . . . . . 8
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((Scalar‘𝑡) = 𝑤 ↔ 𝐿 = 𝐾)) |
17 | 15 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = (Base‘𝐾)) |
18 | | islmhm.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐾) |
19 | 17, 18 | syl6eqr 2662 |
. . . . . . . . 9
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = 𝐵) |
20 | 11 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = (Base‘𝑆)) |
21 | | islmhm.e |
. . . . . . . . . . 11
⊢ 𝐸 = (Base‘𝑆) |
22 | 20, 21 | syl6eqr 2662 |
. . . . . . . . . 10
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = 𝐸) |
23 | 11 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (
·𝑠 ‘𝑠) = ( ·𝑠
‘𝑆)) |
24 | | islmhm.m |
. . . . . . . . . . . . . 14
⊢ · = (
·𝑠 ‘𝑆) |
25 | 23, 24 | syl6eqr 2662 |
. . . . . . . . . . . . 13
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (
·𝑠 ‘𝑠) = · ) |
26 | 25 | oveqd 6566 |
. . . . . . . . . . . 12
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠
‘𝑠)𝑦) = (𝑥 · 𝑦)) |
27 | 26 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑓‘(𝑥 · 𝑦))) |
28 | 6 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (
·𝑠 ‘𝑡) = ( ·𝑠
‘𝑇)) |
29 | | islmhm.n |
. . . . . . . . . . . . 13
⊢ × = (
·𝑠 ‘𝑇) |
30 | 28, 29 | syl6eqr 2662 |
. . . . . . . . . . . 12
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (
·𝑠 ‘𝑡) = × ) |
31 | 30 | oveqd 6566 |
. . . . . . . . . . 11
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)) = (𝑥 × (𝑓‘𝑦))) |
32 | 27, 31 | eqeq12d 2625 |
. . . . . . . . . 10
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))) |
33 | 22, 32 | raleqbidv 3129 |
. . . . . . . . 9
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))) |
34 | 19, 33 | raleqbidv 3129 |
. . . . . . . 8
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))) |
35 | 16, 34 | anbi12d 743 |
. . . . . . 7
⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))))) |
36 | 5, 35 | sbcied 3439 |
. . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ([(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))))) |
37 | 3, 36 | rabeqbidv 3168 |
. . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)))} = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))}) |
38 | | ovex 6577 |
. . . . . 6
⊢ (𝑆 GrpHom 𝑇) ∈ V |
39 | 38 | rabex 4740 |
. . . . 5
⊢ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ∈ V |
40 | 37, 1, 39 | ovmpt2a 6689 |
. . . 4
⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 LMHom 𝑇) = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))}) |
41 | 40 | eleq2d 2673 |
. . 3
⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))})) |
42 | | fveq1 6102 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦))) |
43 | | fveq1 6102 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) |
44 | 43 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (𝑥 × (𝑓‘𝑦)) = (𝑥 × (𝐹‘𝑦))) |
45 | 42, 44 | eqeq12d 2625 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))) |
46 | 45 | 2ralbidv 2972 |
. . . . . 6
⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))) |
47 | 46 | anbi2d 736 |
. . . . 5
⊢ (𝑓 = 𝐹 → ((𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
48 | 47 | elrab 3331 |
. . . 4
⊢ (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
49 | | 3anass 1035 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
50 | 48, 49 | bitr4i 266 |
. . 3
⊢ (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))) |
51 | 41, 50 | syl6bb 275 |
. 2
⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
52 | 2, 51 | biadan2 672 |
1
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |