Step | Hyp | Ref
| Expression |
1 | | hgmapfval.k |
. 2
⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) |
2 | | hgmapfval.i |
. . . 4
⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) |
3 | | hgmapval.h |
. . . . . 6
⊢ 𝐻 = (LHyp‘𝐾) |
4 | 3 | hgmapffval 36195 |
. . . . 5
⊢ (𝐾 ∈ 𝑌 → (HGMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))})) |
5 | 4 | fveq1d 6105 |
. . . 4
⊢ (𝐾 ∈ 𝑌 → ((HGMap‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))})‘𝑊)) |
6 | 2, 5 | syl5eq 2656 |
. . 3
⊢ (𝐾 ∈ 𝑌 → 𝐼 = ((𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))})‘𝑊)) |
7 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊)) |
8 | | hgmapfval.u |
. . . . . . . 8
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
9 | 7, 8 | syl6eqr 2662 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈) |
10 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊)) |
11 | | hgmapfval.m |
. . . . . . . . . 10
⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) |
12 | 10, 11 | syl6eqr 2662 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀) |
13 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑊 → ((LCDual‘𝐾)‘𝑤) = ((LCDual‘𝐾)‘𝑊)) |
14 | 13 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑊 → (
·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠
‘((LCDual‘𝐾)‘𝑊))) |
15 | 14 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑊 → (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣))) |
16 | 15 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑊 → ((𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) ↔ (𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) |
17 | 16 | ralbidv 2969 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 → (∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) |
18 | 17 | riotabidv 6513 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))) = (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) |
19 | 18 | mpteq2dv 4673 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) = (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣))))) |
20 | 19 | eleq2d 2673 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) ↔ 𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))))) |
21 | 12, 20 | sbceqbid 3409 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ([((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) ↔ [𝑀 / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))))) |
22 | 21 | sbcbidv 3457 |
. . . . . . 7
⊢ (𝑤 = 𝑊 →
([(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) ↔
[(Base‘(Scalar‘𝑢)) / 𝑏][𝑀 / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))))) |
23 | 9, 22 | sbceqbid 3409 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ([((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) ↔ [𝑈 / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][𝑀 / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))))) |
24 | | fvex 6113 |
. . . . . . . 8
⊢
((DVecH‘𝐾)‘𝑊) ∈ V |
25 | 8, 24 | eqeltri 2684 |
. . . . . . 7
⊢ 𝑈 ∈ V |
26 | | fvex 6113 |
. . . . . . 7
⊢
(Base‘(Scalar‘𝑢)) ∈ V |
27 | | fvex 6113 |
. . . . . . . 8
⊢
((HDMap‘𝐾)‘𝑊) ∈ V |
28 | 11, 27 | eqeltri 2684 |
. . . . . . 7
⊢ 𝑀 ∈ V |
29 | | simp2 1055 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢))) |
30 | | simp1 1054 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈) |
31 | 30 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈)) |
32 | | hgmapfval.r |
. . . . . . . . . . . 12
⊢ 𝑅 = (Scalar‘𝑈) |
33 | 31, 32 | syl6eqr 2662 |
. . . . . . . . . . 11
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅) |
34 | 33 | fveq2d 6107 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅)) |
35 | 29, 34 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅)) |
36 | | hgmapfval.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑅) |
37 | 35, 36 | syl6eqr 2662 |
. . . . . . . 8
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵) |
38 | | simp2 1055 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵) |
39 | | simp1 1054 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈) |
40 | 39 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈)) |
41 | | hgmapfval.v |
. . . . . . . . . . . . 13
⊢ 𝑉 = (Base‘𝑈) |
42 | 40, 41 | syl6eqr 2662 |
. . . . . . . . . . . 12
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = 𝑉) |
43 | | simp3 1056 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀) |
44 | 39 | fveq2d 6107 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘𝑢) = ( ·𝑠
‘𝑈)) |
45 | | hgmapfval.t |
. . . . . . . . . . . . . . . 16
⊢ · = (
·𝑠 ‘𝑈) |
46 | 44, 45 | syl6eqr 2662 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘𝑢) = · ) |
47 | 46 | oveqd 6566 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑥( ·𝑠
‘𝑢)𝑣) = (𝑥 · 𝑣)) |
48 | 43, 47 | fveq12d 6109 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣))) |
49 | | eqidd 2611 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)) |
50 | | hgmapfval.c |
. . . . . . . . . . . . . . . . 17
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
51 | 49, 50 | syl6eqr 2662 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶) |
52 | 51 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠
‘𝐶)) |
53 | | hgmapfval.s |
. . . . . . . . . . . . . . 15
⊢ ∙ = (
·𝑠 ‘𝐶) |
54 | 52, 53 | syl6eqr 2662 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ∙ ) |
55 | | eqidd 2611 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑦 = 𝑦) |
56 | 43 | fveq1d 6105 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘𝑣) = (𝑀‘𝑣)) |
57 | 54, 55, 56 | oveq123d 6570 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) |
58 | 48, 57 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) ↔ (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
59 | 42, 58 | raleqbidv 3129 |
. . . . . . . . . . 11
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
60 | 38, 59 | riotaeqbidv 6514 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣))) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) |
61 | 38, 60 | mpteq12dv 4663 |
. . . . . . . . 9
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
62 | 61 | eleq2d 2673 |
. . . . . . . 8
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) ↔ 𝑎 ∈ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))) |
63 | 37, 62 | syld3an2 1365 |
. . . . . . 7
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) ↔ 𝑎 ∈ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))) |
64 | 25, 26, 28, 63 | sbc3ie 3474 |
. . . . . 6
⊢
([𝑈 / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][𝑀 / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))) ↔ 𝑎 ∈ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
65 | 23, 64 | syl6bb 275 |
. . . . 5
⊢ (𝑤 = 𝑊 → ([((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) ↔ 𝑎 ∈ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))) |
66 | 65 | abbi1dv 2730 |
. . . 4
⊢ (𝑤 = 𝑊 → {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))} = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
67 | | eqid 2610 |
. . . 4
⊢ (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))}) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))}) |
68 | | fvex 6113 |
. . . . . 6
⊢
(Base‘𝑅)
∈ V |
69 | 36, 68 | eqeltri 2684 |
. . . . 5
⊢ 𝐵 ∈ V |
70 | 69 | mptex 6390 |
. . . 4
⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) ∈ V |
71 | 66, 67, 70 | fvmpt 6191 |
. . 3
⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))})‘𝑊) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
72 | 6, 71 | sylan9eq 2664 |
. 2
⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
73 | 1, 72 | syl 17 |
1
⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |