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Theorem hgmapfval 40349
Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
Hypotheses
Ref Expression
hgmapval.h 𝐻 = (LHyp‘𝐾)
hgmapfval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmapfval.v 𝑉 = (Base‘𝑈)
hgmapfval.t · = ( ·𝑠𝑈)
hgmapfval.r 𝑅 = (Scalar‘𝑈)
hgmapfval.b 𝐵 = (Base‘𝑅)
hgmapfval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hgmapfval.s = ( ·𝑠𝐶)
hgmapfval.m 𝑀 = ((HDMap‘𝐾)‘𝑊)
hgmapfval.i 𝐼 = ((HGMap‘𝐾)‘𝑊)
hgmapfval.k (𝜑 → (𝐾𝑌𝑊𝐻))
Assertion
Ref Expression
hgmapfval (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
Distinct variable groups:   𝑥,𝑣,𝑦,𝐾   𝑣,𝐵,𝑥,𝑦   𝑣,𝑀,𝑥,𝑦   𝑣,𝑈,𝑥,𝑦   𝑣,𝑉   𝑣,𝑊,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣)   𝐶(𝑥,𝑦,𝑣)   𝑅(𝑥,𝑦,𝑣)   (𝑥,𝑦,𝑣)   · (𝑥,𝑦,𝑣)   𝐻(𝑥,𝑦,𝑣)   𝐼(𝑥,𝑦,𝑣)   𝑉(𝑥,𝑦)   𝑌(𝑥,𝑦,𝑣)

Proof of Theorem hgmapfval
Dummy variables 𝑤 𝑎 𝑏 𝑚 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hgmapfval.k . 2 (𝜑 → (𝐾𝑌𝑊𝐻))
2 hgmapfval.i . . . 4 𝐼 = ((HGMap‘𝐾)‘𝑊)
3 hgmapval.h . . . . . 6 𝐻 = (LHyp‘𝐾)
43hgmapffval 40348 . . . . 5 (𝐾𝑌 → (HGMap‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))}))
54fveq1d 6844 . . . 4 (𝐾𝑌 → ((HGMap‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))})‘𝑊))
62, 5eqtrid 2788 . . 3 (𝐾𝑌𝐼 = ((𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))})‘𝑊))
7 fveq2 6842 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 hgmapfval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8eqtr4di 2794 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10 fveq2 6842 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11 hgmapfval.m . . . . . . . . . 10 𝑀 = ((HDMap‘𝐾)‘𝑊)
1210, 11eqtr4di 2794 . . . . . . . . 9 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13 2fveq3 6847 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)))
1413oveqd 7374 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))
1514eqeq2d 2747 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) ↔ (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))
1615ralbidv 3174 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))
1716riotabidv 7315 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))) = (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))
1817mpteq2dv 5207 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)))) = (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))))
1918eleq2d 2823 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)))) ↔ 𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))))
2012, 19sbceqbid 3746 . . . . . . . 8 (𝑤 = 𝑊 → ([((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)))) ↔ [𝑀 / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))))
2120sbcbidv 3798 . . . . . . 7 (𝑤 = 𝑊 → ([(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)))) ↔ [(Base‘(Scalar‘𝑢)) / 𝑏][𝑀 / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))))
229, 21sbceqbid 3746 . . . . . 6 (𝑤 = 𝑊 → ([((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)))) ↔ [𝑈 / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][𝑀 / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))))
238fvexi 6856 . . . . . . 7 𝑈 ∈ V
24 fvex 6855 . . . . . . 7 (Base‘(Scalar‘𝑢)) ∈ V
2511fvexi 6856 . . . . . . 7 𝑀 ∈ V
26 simp2 1137 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢)))
27 simp1 1136 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
2827fveq2d 6846 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
29 hgmapfval.r . . . . . . . . . . . 12 𝑅 = (Scalar‘𝑈)
3028, 29eqtr4di 2794 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
3130fveq2d 6846 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
3226, 31eqtrd 2776 . . . . . . . . 9 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
33 hgmapfval.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3432, 33eqtr4di 2794 . . . . . . . 8 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
35 simp2 1137 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑏 = 𝐵)
36 simp1 1136 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑢 = 𝑈)
3736fveq2d 6846 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
38 hgmapfval.v . . . . . . . . . . . . 13 𝑉 = (Base‘𝑈)
3937, 38eqtr4di 2794 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = 𝑉)
40 simp3 1138 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑚 = 𝑀)
4136fveq2d 6846 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
42 hgmapfval.t . . . . . . . . . . . . . . . 16 · = ( ·𝑠𝑈)
4341, 42eqtr4di 2794 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = · )
4443oveqd 7374 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑥( ·𝑠𝑢)𝑣) = (𝑥 · 𝑣))
4540, 44fveq12d 6849 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
46 eqidd 2737 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊))
47 hgmapfval.c . . . . . . . . . . . . . . . . 17 𝐶 = ((LCDual‘𝐾)‘𝑊)
4846, 47eqtr4di 2794 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
4948fveq2d 6846 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠𝐶))
50 hgmapfval.s . . . . . . . . . . . . . . 15 = ( ·𝑠𝐶)
5149, 50eqtr4di 2794 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = )
52 eqidd 2737 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑦 = 𝑦)
5340fveq1d 6844 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚𝑣) = (𝑀𝑣))
5451, 52, 53oveq123d 7378 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) = (𝑦 (𝑀𝑣)))
5545, 54eqeq12d 2752 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) ↔ (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
5639, 55raleqbidv 3319 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) ↔ ∀𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
5735, 56riotaeqbidv 7316 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
5835, 57mpteq12dv 5196 . . . . . . . . 9 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))) = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
5958eleq2d 2823 . . . . . . . 8 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))) ↔ 𝑎 ∈ (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))))
6034, 59syld3an2 1411 . . . . . . 7 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))) ↔ 𝑎 ∈ (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))))
6123, 24, 25, 60sbc3ie 3825 . . . . . 6 ([𝑈 / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][𝑀 / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))) ↔ 𝑎 ∈ (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
6222, 61bitrdi 286 . . . . 5 (𝑤 = 𝑊 → ([((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)))) ↔ 𝑎 ∈ (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))))
6362abbi1dv 2872 . . . 4 (𝑤 = 𝑊 → {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))} = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
64 eqid 2736 . . . 4 (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))}) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))})
6563, 64, 33mptfvmpt 7178 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))})‘𝑊) = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
666, 65sylan9eq 2796 . 2 ((𝐾𝑌𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
671, 66syl 17 1 (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  [wsbc 3739  cmpt 5188  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  LHypclh 38447  DVecHcdvh 39541  LCDualclcd 40049  HDMapchdma 40255  HGMapchg 40346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-hgmap 40347
This theorem is referenced by:  hgmapval  40350  hgmapfnN  40351
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