Step | Hyp | Ref
| Expression |
1 | | ldualset.w |
. 2
⊢ (𝜑 → 𝑊 ∈ 𝑋) |
2 | | elex 3461 |
. 2
⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) |
3 | | ldualset.d |
. . 3
⊢ 𝐷 = (LDual‘𝑊) |
4 | | fveq2 6839 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊)) |
5 | | ldualset.f |
. . . . . . . 8
⊢ 𝐹 = (LFnl‘𝑊) |
6 | 4, 5 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹) |
7 | 6 | opeq2d 4835 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx),
(LFnl‘𝑤)⟩ =
⟨(Base‘ndx), 𝐹⟩) |
8 | | fveq2 6839 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) |
9 | | ldualset.r |
. . . . . . . . . . . . 13
⊢ 𝑅 = (Scalar‘𝑊) |
10 | 8, 9 | eqtr4di 2795 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅) |
11 | 10 | fveq2d 6843 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 →
(+g‘(Scalar‘𝑤)) = (+g‘𝑅)) |
12 | | ldualset.a |
. . . . . . . . . . 11
⊢ + =
(+g‘𝑅) |
13 | 11, 12 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 →
(+g‘(Scalar‘𝑤)) = + ) |
14 | 13 | ofeqd 7611 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ∘f
(+g‘(Scalar‘𝑤)) = ∘f + ) |
15 | 6 | sqxpeqd 5663 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹)) |
16 | 14, 15 | reseq12d 5936 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ( ∘f
(+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘f + ↾
(𝐹 × 𝐹))) |
17 | | ldualset.p |
. . . . . . . 8
⊢ ✚ = (
∘f + ↾ (𝐹 × 𝐹)) |
18 | 16, 17 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ( ∘f
(+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ✚ ) |
19 | 18 | opeq2d 4835 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), (
∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx),
✚
⟩) |
20 | 10 | fveq2d 6843 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 →
(oppr‘(Scalar‘𝑤)) = (oppr‘𝑅)) |
21 | | ldualset.o |
. . . . . . . 8
⊢ 𝑂 =
(oppr‘𝑅) |
22 | 20, 21 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑤 = 𝑊 →
(oppr‘(Scalar‘𝑤)) = 𝑂) |
23 | 22 | opeq2d 4835 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx),
(oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩) |
24 | 7, 19, 23 | tpeq123d 4707 |
. . . . 5
⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx),
(LFnl‘𝑤)⟩,
⟨(+g‘ndx), ( ∘f
(+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx),
(oppr‘(Scalar‘𝑤))⟩} = {⟨(Base‘ndx), 𝐹⟩,
⟨(+g‘ndx), ✚ ⟩,
⟨(Scalar‘ndx), 𝑂⟩}) |
25 | 10 | fveq2d 6843 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅)) |
26 | | ldualset.k |
. . . . . . . . . 10
⊢ 𝐾 = (Base‘𝑅) |
27 | 25, 26 | eqtr4di 2795 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾) |
28 | 10 | fveq2d 6843 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 →
(.r‘(Scalar‘𝑤)) = (.r‘𝑅)) |
29 | | ldualset.t |
. . . . . . . . . . . 12
⊢ · =
(.r‘𝑅) |
30 | 28, 29 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 →
(.r‘(Scalar‘𝑤)) = · ) |
31 | 30 | ofeqd 7611 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ∘f
(.r‘(Scalar‘𝑤)) = ∘f · ) |
32 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓) |
33 | | fveq2 6839 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) |
34 | | ldualset.v |
. . . . . . . . . . . 12
⊢ 𝑉 = (Base‘𝑊) |
35 | 33, 34 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉) |
36 | 35 | xpeq1d 5660 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘})) |
37 | 31, 32, 36 | oveq123d 7372 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘f · (𝑉 × {𝑘}))) |
38 | 27, 6, 37 | mpoeq123dv 7426 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f · (𝑉 × {𝑘})))) |
39 | | ldualset.s |
. . . . . . . 8
⊢ ∙ =
(𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f · (𝑉 × {𝑘}))) |
40 | 38, 39 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = ∙ ) |
41 | 40 | opeq2d 4835 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ⟨(
·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨(
·𝑠 ‘ndx), ∙
⟩) |
42 | 41 | sneqd 4596 |
. . . . 5
⊢ (𝑤 = 𝑊 → {⟨(
·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨(
·𝑠 ‘ndx), ∙
⟩}) |
43 | 24, 42 | uneq12d 4122 |
. . . 4
⊢ (𝑤 = 𝑊 → ({⟨(Base‘ndx),
(LFnl‘𝑤)⟩,
⟨(+g‘ndx), ( ∘f
(+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx),
(oppr‘(Scalar‘𝑤))⟩} ∪ {⟨(
·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩}) = ({⟨(Base‘ndx),
𝐹⟩,
⟨(+g‘ndx), ✚ ⟩,
⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨(
·𝑠 ‘ndx), ∙
⟩})) |
44 | | df-ldual 37524 |
. . . 4
⊢ LDual =
(𝑤 ∈ V ↦
({⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), (
∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx),
(oppr‘(Scalar‘𝑤))⟩} ∪ {⟨(
·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩})) |
45 | | tpex 7673 |
. . . . 5
⊢
{⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx),
✚
⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V |
46 | | snex 5386 |
. . . . 5
⊢ {⟨(
·𝑠 ‘ndx), ∙ ⟩} ∈
V |
47 | 45, 46 | unex 7672 |
. . . 4
⊢
({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx),
✚
⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨(
·𝑠 ‘ndx), ∙ ⟩}) ∈
V |
48 | 43, 44, 47 | fvmpt 6945 |
. . 3
⊢ (𝑊 ∈ V →
(LDual‘𝑊) =
({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx),
✚
⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨(
·𝑠 ‘ndx), ∙
⟩})) |
49 | 3, 48 | eqtrid 2789 |
. 2
⊢ (𝑊 ∈ V → 𝐷 = ({⟨(Base‘ndx),
𝐹⟩,
⟨(+g‘ndx), ✚ ⟩,
⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨(
·𝑠 ‘ndx), ∙
⟩})) |
50 | 1, 2, 49 | 3syl 18 |
1
⊢ (𝜑 → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩,
⟨(+g‘ndx), ✚ ⟩,
⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨(
·𝑠 ‘ndx), ∙
⟩})) |