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Theorem hgmapfval 35996
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 35995 . . . . 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 6087 . . . 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 2652 . . 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 6085 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 hgmapfval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8syl6eqr 2658 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10 fveq2 6085 . . . . . . . . . 10 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11 hgmapfval.m . . . . . . . . . 10 𝑀 = ((HDMap‘𝐾)‘𝑊)
1210, 11syl6eqr 2658 . . . . . . . . 9 (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13 fveq2 6085 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → ((LCDual‘𝐾)‘𝑤) = ((LCDual‘𝐾)‘𝑊))
1413fveq2d 6089 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)))
1514oveqd 6541 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)))
1615eqeq2d 2616 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣)) ↔ (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))))
1716ralbidv 2965 . . . . . . . . . . . 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))))
1817riotabidv 6488 . . . . . . . . . . 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))))
1918mpteq2dv 4664 . . . . . . . . . 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)))))
2019eleq2d 2669 . . . . . . . . 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))))))
2112, 20sbceqbid 3405 . . . . . . . 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))))))
2221sbcbidv 3453 . . . . . . 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))))))
239, 22sbceqbid 3405 . . . . . 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))))))
24 fvex 6095 . . . . . . . 8 ((DVecH‘𝐾)‘𝑊) ∈ V
258, 24eqeltri 2680 . . . . . . 7 𝑈 ∈ V
26 fvex 6095 . . . . . . 7 (Base‘(Scalar‘𝑢)) ∈ V
27 fvex 6095 . . . . . . . 8 ((HDMap‘𝐾)‘𝑊) ∈ V
2811, 27eqeltri 2680 . . . . . . 7 𝑀 ∈ V
29 simp2 1054 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢)))
30 simp1 1053 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
3130fveq2d 6089 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
32 hgmapfval.r . . . . . . . . . . . 12 𝑅 = (Scalar‘𝑈)
3331, 32syl6eqr 2658 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
3433fveq2d 6089 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
3529, 34eqtrd 2640 . . . . . . . . 9 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
36 hgmapfval.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3735, 36syl6eqr 2658 . . . . . . . 8 ((𝑢 = 𝑈𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
38 simp2 1054 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑏 = 𝐵)
39 simp1 1053 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑢 = 𝑈)
4039fveq2d 6089 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
41 hgmapfval.v . . . . . . . . . . . . 13 𝑉 = (Base‘𝑈)
4240, 41syl6eqr 2658 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (Base‘𝑢) = 𝑉)
43 simp3 1055 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑚 = 𝑀)
4439fveq2d 6089 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = ( ·𝑠𝑈))
45 hgmapfval.t . . . . . . . . . . . . . . . 16 · = ( ·𝑠𝑈)
4644, 45syl6eqr 2658 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠𝑢) = · )
4746oveqd 6541 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑥( ·𝑠𝑢)𝑣) = (𝑥 · 𝑣))
4843, 47fveq12d 6091 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
49 eqidd 2607 . . . . . . . . . . . . . . . . 17 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊))
50 hgmapfval.c . . . . . . . . . . . . . . . . 17 𝐶 = ((LCDual‘𝐾)‘𝑊)
5149, 50syl6eqr 2658 . . . . . . . . . . . . . . . 16 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
5251fveq2d 6089 . . . . . . . . . . . . . . 15 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠𝐶))
53 hgmapfval.s . . . . . . . . . . . . . . 15 = ( ·𝑠𝐶)
5452, 53syl6eqr 2658 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = )
55 eqidd 2607 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → 𝑦 = 𝑦)
5643fveq1d 6087 . . . . . . . . . . . . . 14 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑚𝑣) = (𝑀𝑣))
5754, 55, 56oveq123d 6545 . . . . . . . . . . . . 13 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) = (𝑦 (𝑀𝑣)))
5848, 57eqeq12d 2621 . . . . . . . . . . . 12 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → ((𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) ↔ (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
5942, 58raleqbidv 3125 . . . . . . . . . . 11 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣)) ↔ ∀𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
6038, 59riotaeqbidv 6489 . . . . . . . . . 10 ((𝑢 = 𝑈𝑏 = 𝐵𝑚 = 𝑀) → (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚𝑣))) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
6138, 60mpteq12dv 4654 . . . . . . . . 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)))))
6261eleq2d 2669 . . . . . . . 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))))))
6337, 62syld3an2 1364 . . . . . . 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))))))
6425, 26, 28, 63sbc3ie 3470 . . . . . 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)))))
6523, 64syl6bb 274 . . . . 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))))))
6665abbi1dv 2726 . . . 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)))))
67 eqid 2606 . . . 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))))})
68 fvex 6095 . . . . . 6 (Base‘𝑅) ∈ V
6936, 68eqeltri 2680 . . . . 5 𝐵 ∈ V
7069mptex 6365 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))) ∈ V
7166, 67, 70fvmpt 6173 . . 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)))))
726, 71sylan9eq 2660 . 2 ((𝐾𝑌𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
731, 72syl 17 1 (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2592  wral 2892  Vcvv 3169  [wsbc 3398  cmpt 4634  cfv 5787  crio 6485  (class class class)co 6524  Basecbs 15638  Scalarcsca 15714   ·𝑠 cvsca 15715  LHypclh 34088  DVecHcdvh 35185  LCDualclcd 35693  HDMapchdma 35900  HGMapchg 35993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-hgmap 35994
This theorem is referenced by:  hgmapval  35997  hgmapfnN  35998
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