Theorem hgmapfval 42522

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 42521	. . . . 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 6873	. . . 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 2812	. . 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 6871	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		hgmapfval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11		hgmapfval.m	. . . . . . . . . 10 ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13		2fveq3 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)))
14	13	oveqd 7417	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))
15	14	eqeq2d 2776	. . . . . . . . . . . . 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 3188	. . . . . . . . . . . 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 7359	. . . . . . . . . . 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 5199	. . . . . . . . . 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 2851	. . . . . . . . 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 3754	. . . . . . . 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 3802	. . . . . . 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 3754	. . . . . 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 6885	. . . . . . 7 ⊢ 𝑈 ∈ V
24		fvex 6884	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑢)) ∈ V
25	11	fvexi 6885	. . . . . . 7 ⊢ 𝑀 ∈ V
26		simp2 1153	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢)))
27		simp1 1152	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
28	27	fveq2d 6875	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
29		hgmapfval.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈)
30	28, 29	eqtr4di 2818	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
31	30	fveq2d 6875	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
32	26, 31	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
33		hgmapfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
34	32, 33	eqtr4di 2818	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
35		simp2 1153	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
36		simp1 1152	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
37	36	fveq2d 6875	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
38		hgmapfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (Base‘𝑈)
39	37, 38	eqtr4di 2818	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = 𝑉)
40		simp3 1154	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
41	36	fveq2d 6875	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘𝑢) = ( ·𝑠 ‘𝑈))
42		hgmapfval.t	. . . . . . . . . . . . . . . 16 ⊢ · = ( ·𝑠 ‘𝑈)
43	41, 42	eqtr4di 2818	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘𝑢) = · )
44	43	oveqd 7417	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑥( ·𝑠 ‘𝑢)𝑣) = (𝑥 · 𝑣))
45	40, 44	fveq12d 6878	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
46		eqidd 2766	. . . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
49	48	fveq2d 6875	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘𝐶))
50		hgmapfval.s	. . . . . . . . . . . . . . 15 ⊢ ∙ = ( ·𝑠 ‘𝐶)
51	49, 50	eqtr4di 2818	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ∙ )
52		eqidd 2766	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑦 = 𝑦)
53	40	fveq1d 6873	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘𝑣) = (𝑀‘𝑣))
54	51, 52, 53	oveq123d 7421	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))
55	45, 54	eqeq12d 2781	. . . . . . . . . . . 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 3339	. . . . . . . . . . 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 7360	. . . . . . . . . 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 5192	. . . . . . . . 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 2851	. . . . . . . 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 1434	. . . . . . 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 3824	. . . . . 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 290	. . . . 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 2899	. . . 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 2765	. . . 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 7216	. . 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 2820	. 2 ⊢ ((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))
67	1, 66	syl 18	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  [wsbc 3747   ↦ cmpt 5186  ‘cfv 6525  ℩crio 7356  (class class class)co 7400  Basecbs 17259  Scalarcsca 17303   ·_𝑠 cvsca 17304  LHypclh 40620  DVecHcdvh 41714  LCDualclcd 42222  HDMapchdma 42428  HGMapchg 42519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-hgmap 42520

This theorem is referenced by:  hgmapval  42523  hgmapfnN  42524