| Step | Hyp | Ref
| Expression |
| 1 | | ldualset.w |
. 2
⊢ (𝜑 → 𝑊 ∈ 𝑋) |
| 2 | | elex 3501 |
. 2
⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) |
| 3 | | ldualset.d |
. . 3
⊢ 𝐷 = (LDual‘𝑊) |
| 4 | | fveq2 6906 |
. . . . . . . 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 4880 |
. . . . . 6
⊢ (𝑤 = 𝑊 → 〈(Base‘ndx),
(LFnl‘𝑤)〉 =
〈(Base‘ndx), 𝐹〉) |
| 8 | | fveq2 6906 |
. . . . . . . . . . . . 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 6910 |
. . . . . . . . . . 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 7699 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ∘f
(+g‘(Scalar‘𝑤)) = ∘f + ) |
| 15 | 6 | sqxpeqd 5717 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹)) |
| 16 | 14, 15 | reseq12d 5998 |
. . . . . . . 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 4880 |
. . . . . 6
⊢ (𝑤 = 𝑊 → 〈(+g‘ndx), (
∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))〉 = 〈(+g‘ndx),
✚
〉) |
| 20 | 10 | fveq2d 6910 |
. . . . . . . 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 4880 |
. . . . . 6
⊢ (𝑤 = 𝑊 → 〈(Scalar‘ndx),
(oppr‘(Scalar‘𝑤))〉 = 〈(Scalar‘ndx), 𝑂〉) |
| 24 | 7, 19, 23 | tpeq123d 4748 |
. . . . 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 6910 |
. . . . . . . . . 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 6910 |
. . . . . . . . . . . 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 7699 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ∘f
(.r‘(Scalar‘𝑤)) = ∘f · ) |
| 32 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓) |
| 33 | | fveq2 6906 |
. . . . . . . . . . . 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 5714 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘})) |
| 37 | 31, 32, 36 | oveq123d 7452 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (𝑓 ∘f
(.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘f · (𝑉 × {𝑘}))) |
| 38 | 27, 6, 37 | mpoeq123dv 7508 |
. . . . . . . 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 4880 |
. . . . . 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 4638 |
. . . . 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 4169 |
. . . 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 39125 |
. . . 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 7766 |
. . . . 5
⊢
{〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx),
✚
〉, 〈(Scalar‘ndx), 𝑂〉} ∈ V |
| 46 | | snex 5436 |
. . . . 5
⊢ {〈(
·𝑠 ‘ndx), ∙ 〉} ∈
V |
| 47 | 45, 46 | unex 7764 |
. . . 4
⊢
({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx),
✚
〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈(
·𝑠 ‘ndx), ∙ 〉}) ∈
V |
| 48 | 43, 44, 47 | fvmpt 7016 |
. . 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), ∙
〉})) |