| Step | Hyp | Ref
| Expression |
| 1 | | elex 3501 |
. 2
⊢ (𝐾 ∈ 𝑋 → 𝐾 ∈ V) |
| 2 | | fveq2 6906 |
. . . . 5
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) |
| 3 | | hgmapval.h |
. . . . 5
⊢ 𝐻 = (LHyp‘𝐾) |
| 4 | 2, 3 | eqtr4di 2795 |
. . . 4
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
| 5 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾)) |
| 6 | 5 | fveq1d 6908 |
. . . . . 6
⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤)) |
| 7 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (HDMap‘𝑘) = (HDMap‘𝐾)) |
| 8 | 7 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → ((HDMap‘𝑘)‘𝑤) = ((HDMap‘𝐾)‘𝑤)) |
| 9 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝐾 → (LCDual‘𝑘) = (LCDual‘𝐾)) |
| 10 | 9 | fveq1d 6908 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝐾 → ((LCDual‘𝑘)‘𝑤) = ((LCDual‘𝐾)‘𝑤)) |
| 11 | 10 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝐾 → (
·𝑠 ‘((LCDual‘𝑘)‘𝑤)) = ( ·𝑠
‘((LCDual‘𝐾)‘𝑤))) |
| 12 | 11 | oveqd 7448 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝐾 → (𝑦( ·𝑠
‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))) |
| 13 | 12 | eqeq2d 2748 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → ((𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣)) ↔ (𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))) |
| 14 | 13 | ralbidv 3178 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (∀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)))) |
| 15 | 14 | riotabidv 7390 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐾 → (℩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)))) |
| 16 | 15 | mpteq2dv 5244 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (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))))) |
| 17 | 16 | eleq2d 2827 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (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)))))) |
| 18 | 8, 17 | sbceqbid 3795 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → ([((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣)))) ↔ [((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))))) |
| 19 | 18 | sbcbidv 3845 |
. . . . . 6
⊢ (𝑘 = 𝐾 →
([(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][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)))))) |
| 20 | 6, 19 | sbceqbid 3795 |
. . . . 5
⊢ (𝑘 = 𝐾 → ([((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)))) ↔ [((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)))))) |
| 21 | 20 | abbidv 2808 |
. . . 4
⊢ (𝑘 = 𝐾 → {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))))} = {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))))}) |
| 22 | 4, 21 | mpteq12dv 5233 |
. . 3
⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {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))))})) |
| 23 | | df-hgmap 41886 |
. . 3
⊢ HGMap =
(𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {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))))})) |
| 24 | 22, 23, 3 | mptfvmpt 7248 |
. 2
⊢ (𝐾 ∈ V →
(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))))})) |
| 25 | 1, 24 | syl 17 |
1
⊢ (𝐾 ∈ 𝑋 → (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))))})) |