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Theorem hgmapfval 38902
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 38901 . . . . 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 6665 . . . 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, 5syl5eq 2865 . . 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 6663 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 hgmapfval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8syl6eqr 2871 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10 fveq2 6663 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11 hgmapfval.m . . . . . . . . . 10 𝑀 = ((HDMap‘𝐾)‘𝑊)
1210, 11syl6eqr 2871 . . . . . . . . 9 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13 2fveq3 6668 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)))
1413oveqd 7162 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))
1514eqeq2d 2829 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) ↔ (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))
1615ralbidv 3194 . . . . . . . . . . . 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 7105 . . . . . . . . . . 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 5153 . . . . . . . . . 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 2895 . . . . . . . . 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 3776 . . . . . . . 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 3824 . . . . . . 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 3776 . . . . . 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 6677 . . . . . . 7 𝑈 ∈ V
24 fvex 6676 . . . . . . 7 (Base‘(Scalar‘𝑢)) ∈ V
2511fvexi 6677 . . . . . . 7 𝑀 ∈ V
26 simp2 1129 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢)))
27 simp1 1128 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
2827fveq2d 6667 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
29 hgmapfval.r . . . . . . . . . . . 12 𝑅 = (Scalar‘𝑈)
3028, 29syl6eqr 2871 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
3130fveq2d 6667 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
3226, 31eqtrd 2853 . . . . . . . . 9 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
33 hgmapfval.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3432, 33syl6eqr 2871 . . . . . . . 8 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
35 simp2 1129 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑏 = 𝐵)
36 simp1 1128 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑢 = 𝑈)
3736fveq2d 6667 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
38 hgmapfval.v . . . . . . . . . . . . 13 𝑉 = (Base‘𝑈)
3937, 38syl6eqr 2871 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = 𝑉)
40 simp3 1130 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑚 = 𝑀)
4136fveq2d 6667 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
42 hgmapfval.t . . . . . . . . . . . . . . . 16 · = ( ·𝑠𝑈)
4341, 42syl6eqr 2871 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = · )
4443oveqd 7162 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑥( ·𝑠𝑢)𝑣) = (𝑥 · 𝑣))
4540, 44fveq12d 6670 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
46 eqidd 2819 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊))
47 hgmapfval.c . . . . . . . . . . . . . . . . 17 𝐶 = ((LCDual‘𝐾)‘𝑊)
4846, 47syl6eqr 2871 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
4948fveq2d 6667 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠𝐶))
50 hgmapfval.s . . . . . . . . . . . . . . 15 = ( ·𝑠𝐶)
5149, 50syl6eqr 2871 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = )
52 eqidd 2819 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑦 = 𝑦)
5340fveq1d 6665 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚𝑣) = (𝑀𝑣))
5451, 52, 53oveq123d 7166 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) = (𝑦 (𝑀𝑣)))
5545, 54eqeq12d 2834 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) ↔ (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
5639, 55raleqbidv 3399 . . . . . . . . . . 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 7106 . . . . . . . . . 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 5142 . . . . . . . . 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 2895 . . . . . . . 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 1403 . . . . . . 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 3849 . . . . . 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, 61syl6bb 288 . . . . 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 2949 . . . 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 2818 . . . 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 6981 . . 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 2873 . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wral 3135  [wsbc 3769  cmpt 5137  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  LHypclh 37000  DVecHcdvh 38094  LCDualclcd 38602  HDMapchdma 38808  HGMapchg 38899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-hgmap 38900
This theorem is referenced by:  hgmapval  38903  hgmapfnN  38904
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