hgmapfval - Mathbox for Norm Megill

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Theorem hgmapfval 40757

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	⊢ · = ( ·_𝑠 ‘𝑈)
hgmapfval.r	⊢ 𝑅 = (Scalar‘𝑈)
hgmapfval.b	⊢ 𝐵 = (Base‘𝑅)
hgmapfval.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hgmapfval.s	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hgmapfval.m	⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)
hgmapfval.i	⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊)
hgmapfval.k	⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
hgmapfval	⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))

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.

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 40756	. . . . 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 6894	. . . 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 2785	. . 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 6892	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		hgmapfval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2791	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11		hgmapfval.m	. . . . . . . . . 10 ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13		2fveq3 6897	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ( ·_𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·_𝑠 ‘((LCDual‘𝐾)‘𝑊)))
14	13	oveqd 7426	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (𝑦( ·_𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) = (𝑦( ·_𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))
15	14	eqeq2d 2744	. . . . . . . . . . . . 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 7367	. . . . . . . . . . 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 5251	. . . . . . . . . 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 2820	. . . . . . . . 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 3785	. . . . . . . 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 3837	. . . . . . 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 3785	. . . . . 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 6906	. . . . . . 7 ⊢ 𝑈 ∈ V
24		fvex 6905	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑢)) ∈ V
25	11	fvexi 6906	. . . . . . 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 6896	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
29		hgmapfval.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈)
30	28, 29	eqtr4di 2791	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
31	30	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
32	26, 31	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
33		hgmapfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
34	32, 33	eqtr4di 2791	. . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
38		hgmapfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (Base‘𝑈)
39	37, 38	eqtr4di 2791	. . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·_𝑠 ‘𝑢) = ( ·_𝑠 ‘𝑈))
42		hgmapfval.t	. . . . . . . . . . . . . . . 16 ⊢ · = ( ·_𝑠 ‘𝑈)
43	41, 42	eqtr4di 2791	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·_𝑠 ‘𝑢) = · )
44	43	oveqd 7426	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑥( ·_𝑠 ‘𝑢)𝑣) = (𝑥 · 𝑣))
45	40, 44	fveq12d 6899	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘(𝑥( ·_𝑠 ‘𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
46		eqidd 2734	. . . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
49	48	fveq2d 6896	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·_𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·_𝑠 ‘𝐶))
50		hgmapfval.s	. . . . . . . . . . . . . . 15 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
51	49, 50	eqtr4di 2791	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·_𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ∙ )
52		eqidd 2734	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑦 = 𝑦)
53	40	fveq1d 6894	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘𝑣) = (𝑀‘𝑣))
54	51, 52, 53	oveq123d 7430	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑦( ·_𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))
55	45, 54	eqeq12d 2749	. . . . . . . . . . . 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 3343	. . . . . . . . . . 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 7368	. . . . . . . . . 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 5240	. . . . . . . . 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 2820	. . . . . . . 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 1412	. . . . . . 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 3864	. . . . . 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 2869	. . . 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 2733	. . . 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 7230	. . 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 2793	. 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)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 [wsbc 3778 ↦ cmpt 5232 ‘cfv 6544 ℩crio 7364 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 ·_𝑠 cvsca 17201 LHypclh 38855 DVecHcdvh 39949 LCDualclcd 40457 HDMapchdma 40663 HGMapchg 40754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-hgmap 40755

This theorem is referenced by: hgmapval 40758 hgmapfnN 40759

W3C validator