| 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 41887 |
. . . . 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 6908 |
. . . 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 | eqtrid 2789 |
. . 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 6906 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊)) |
| 8 | | hgmapfval.u |
. . . . . . . 8
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| 9 | 7, 8 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈) |
| 10 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊)) |
| 11 | | hgmapfval.m |
. . . . . . . . . 10
⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) |
| 12 | 10, 11 | eqtr4di 2795 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀) |
| 13 | | 2fveq3 6911 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑊 → (
·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠
‘((LCDual‘𝐾)‘𝑊))) |
| 14 | 13 | oveqd 7448 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑊 → (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣))) |
| 15 | 14 | eqeq2d 2748 |
. . . . . . . . . . . . 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)))) |
| 16 | 15 | ralbidv 3178 |
. . . . . . . . . . . 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)))) |
| 17 | 16 | riotabidv 7390 |
. . . . . . . . . . 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)))) |
| 18 | 17 | mpteq2dv 5244 |
. . . . . . . . . 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))))) |
| 19 | 18 | eleq2d 2827 |
. . . . . . . . 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)))))) |
| 20 | 12, 19 | sbceqbid 3795 |
. . . . . . . 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)))))) |
| 21 | 20 | sbcbidv 3845 |
. . . . . . 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)))))) |
| 22 | 9, 21 | sbceqbid 3795 |
. . . . . 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)))))) |
| 23 | 8 | fvexi 6920 |
. . . . . . 7
⊢ 𝑈 ∈ V |
| 24 | | fvex 6919 |
. . . . . . 7
⊢
(Base‘(Scalar‘𝑢)) ∈ V |
| 25 | 11 | fvexi 6920 |
. . . . . . 7
⊢ 𝑀 ∈ V |
| 26 | | simp2 1138 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢))) |
| 27 | | simp1 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈) |
| 28 | 27 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈)) |
| 29 | | hgmapfval.r |
. . . . . . . . . . . 12
⊢ 𝑅 = (Scalar‘𝑈) |
| 30 | 28, 29 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅) |
| 31 | 30 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅)) |
| 32 | 26, 31 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅)) |
| 33 | | hgmapfval.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝑅) |
| 34 | 32, 33 | eqtr4di 2795 |
. . . . . . . 8
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵) |
| 35 | | simp2 1138 |
. . . . . . . . . 10
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵) |
| 36 | | simp1 1137 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈) |
| 37 | 36 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈)) |
| 38 | | hgmapfval.v |
. . . . . . . . . . . . 13
⊢ 𝑉 = (Base‘𝑈) |
| 39 | 37, 38 | eqtr4di 2795 |
. . . . . . . . . . . 12
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = 𝑉) |
| 40 | | simp3 1139 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀) |
| 41 | 36 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘𝑢) = ( ·𝑠
‘𝑈)) |
| 42 | | hgmapfval.t |
. . . . . . . . . . . . . . . 16
⊢ · = (
·𝑠 ‘𝑈) |
| 43 | 41, 42 | eqtr4di 2795 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘𝑢) = · ) |
| 44 | 43 | oveqd 7448 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑥( ·𝑠
‘𝑢)𝑣) = (𝑥 · 𝑣)) |
| 45 | 40, 44 | fveq12d 6913 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣))) |
| 46 | | eqidd 2738 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)) |
| 47 | | hgmapfval.c |
. . . . . . . . . . . . . . . . 17
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
| 48 | 46, 47 | eqtr4di 2795 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶) |
| 49 | 48 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠
‘𝐶)) |
| 50 | | hgmapfval.s |
. . . . . . . . . . . . . . 15
⊢ ∙ = (
·𝑠 ‘𝐶) |
| 51 | 49, 50 | eqtr4di 2795 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (
·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ∙ ) |
| 52 | | eqidd 2738 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑦 = 𝑦) |
| 53 | 40 | fveq1d 6908 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘𝑣) = (𝑀‘𝑣)) |
| 54 | 51, 52, 53 | oveq123d 7452 |
. . . . . . . . . . . . 13
⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑦( ·𝑠
‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) |
| 55 | 45, 54 | eqeq12d 2753 |
. . . . . . . . . . . 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)))) |
| 56 | 39, 55 | raleqbidv 3346 |
. . . . . . . . . . 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)))) |
| 57 | 35, 56 | riotaeqbidv 7391 |
. . . . . . . . . 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)))) |
| 58 | 35, 57 | mpteq12dv 5233 |
. . . . . . . . 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))))) |
| 59 | 58 | eleq2d 2827 |
. . . . . . . 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)))))) |
| 60 | 34, 59 | syld3an2 1413 |
. . . . . . 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)))))) |
| 61 | 23, 24, 25, 60 | sbc3ie 3868 |
. . . . . 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))))) |
| 62 | 22, 61 | bitrdi 287 |
. . . . 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)))))) |
| 63 | 62 | eqabcdv 2876 |
. . . 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))))) |
| 64 | | eqid 2737 |
. . . 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))))}) |
| 65 | 63, 64, 33 | mptfvmpt 7248 |
. . 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))))) |
| 66 | 6, 65 | sylan9eq 2797 |
. 2
⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |
| 67 | 1, 66 | syl 17 |
1
⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) |