Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hgmapfval Structured version   Visualization version   GIF version

Theorem hgmapfval 39094
 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 39093 . . . . 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 6661 . . . 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 2871 . . 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 6659 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 hgmapfval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8syl6eqr 2877 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10 fveq2 6659 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11 hgmapfval.m . . . . . . . . . 10 𝑀 = ((HDMap‘𝐾)‘𝑊)
1210, 11syl6eqr 2877 . . . . . . . . 9 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13 2fveq3 6664 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)))
1413oveqd 7163 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))
1514eqeq2d 2835 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) ↔ (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))
1615ralbidv 3192 . . . . . . . . . . . 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 7106 . . . . . . . . . . 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 5149 . . . . . . . . . 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 2901 . . . . . . . . 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 3765 . . . . . . . 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 3812 . . . . . . 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 3765 . . . . . 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 6673 . . . . . . 7 𝑈 ∈ V
24 fvex 6672 . . . . . . 7 (Base‘(Scalar‘𝑢)) ∈ V
2511fvexi 6673 . . . . . . 7 𝑀 ∈ V
26 simp2 1134 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢)))
27 simp1 1133 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
2827fveq2d 6663 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
29 hgmapfval.r . . . . . . . . . . . 12 𝑅 = (Scalar‘𝑈)
3028, 29syl6eqr 2877 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
3130fveq2d 6663 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
3226, 31eqtrd 2859 . . . . . . . . 9 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
33 hgmapfval.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3432, 33syl6eqr 2877 . . . . . . . 8 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
35 simp2 1134 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑏 = 𝐵)
36 simp1 1133 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑢 = 𝑈)
3736fveq2d 6663 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
38 hgmapfval.v . . . . . . . . . . . . 13 𝑉 = (Base‘𝑈)
3937, 38syl6eqr 2877 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = 𝑉)
40 simp3 1135 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑚 = 𝑀)
4136fveq2d 6663 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
42 hgmapfval.t . . . . . . . . . . . . . . . 16 · = ( ·𝑠𝑈)
4341, 42syl6eqr 2877 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = · )
4443oveqd 7163 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑥( ·𝑠𝑢)𝑣) = (𝑥 · 𝑣))
4540, 44fveq12d 6666 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
46 eqidd 2825 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊))
47 hgmapfval.c . . . . . . . . . . . . . . . . 17 𝐶 = ((LCDual‘𝐾)‘𝑊)
4846, 47syl6eqr 2877 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
4948fveq2d 6663 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠𝐶))
50 hgmapfval.s . . . . . . . . . . . . . . 15 = ( ·𝑠𝐶)
5149, 50syl6eqr 2877 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = )
52 eqidd 2825 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑦 = 𝑦)
5340fveq1d 6661 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚𝑣) = (𝑀𝑣))
5451, 52, 53oveq123d 7167 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) = (𝑦 (𝑀𝑣)))
5545, 54eqeq12d 2840 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) ↔ (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
5639, 55raleqbidv 3393 . . . . . . . . . . 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 7107 . . . . . . . . . 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 5138 . . . . . . . . 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 2901 . . . . . . . 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 1408 . . . . . . 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 3836 . . . . . 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 290 . . . . 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 2954 . . . 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 2824 . . . 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 6980 . . 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 2879 . 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  {cab 2802  ∀wral 3133  [wsbc 3758   ↦ cmpt 5133  ‘cfv 6344  ℩crio 7103  (class class class)co 7146  Basecbs 16481  Scalarcsca 16566   ·𝑠 cvsca 16567  LHypclh 37192  DVecHcdvh 38286  LCDualclcd 38794  HDMapchdma 39000  HGMapchg 39091 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-hgmap 39092 This theorem is referenced by:  hgmapval  39095  hgmapfnN  39096
 Copyright terms: Public domain W3C validator