Theorem hgmapfval 42911

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 42910	. . . . 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 6879	. . . 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 2808	. . 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 6877	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		hgmapfval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
11		hgmapfval.m	. . . . . . . . . 10 ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((HDMap‘𝐾)‘𝑤) = 𝑀)
13		2fveq3 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑤)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)))
14	13	oveqd 7429	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)))
15	14	eqeq2d 2772	. . . . . . . . . . . . 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 3186	. . . . . . . . . . . 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 7371	. . . . . . . . . . 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 2847	. . . . . . . . 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 3746	. . . . . . . 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 3794	. . . . . . 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 3746	. . . . . 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 6891	. . . . . . 7 ⊢ 𝑈 ∈ V
24		fvex 6890	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑢)) ∈ V
25	11	fvexi 6891	. . . . . . 7 ⊢ 𝑀 ∈ V
26		simp2 1155	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘(Scalar‘𝑢)))
27		simp1 1154	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
28	27	fveq2d 6881	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = (Scalar‘𝑈))
29		hgmapfval.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑈)
30	28, 29	eqtr4di 2814	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Scalar‘𝑢) = 𝑅)
31	30	fveq2d 6881	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑢)) = (Base‘𝑅))
32	26, 31	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = (Base‘𝑅))
33		hgmapfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
34	32, 33	eqtr4di 2814	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = (Base‘(Scalar‘𝑢)) ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
35		simp2 1155	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑏 = 𝐵)
36		simp1 1154	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑢 = 𝑈)
37	36	fveq2d 6881	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = (Base‘𝑈))
38		hgmapfval.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (Base‘𝑈)
39	37, 38	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (Base‘𝑢) = 𝑉)
40		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀)
41	36	fveq2d 6881	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘𝑢) = ( ·𝑠 ‘𝑈))
42		hgmapfval.t	. . . . . . . . . . . . . . . 16 ⊢ · = ( ·𝑠 ‘𝑈)
43	41, 42	eqtr4di 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘𝑢) = · )
44	43	oveqd 7429	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑥( ·𝑠 ‘𝑢)𝑣) = (𝑥 · 𝑣))
45	40, 44	fveq12d 6884	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑀‘(𝑥 · 𝑣)))
46		eqidd 2762	. . . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ((LCDual‘𝐾)‘𝑊) = 𝐶)
49	48	fveq2d 6881	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘𝐶))
50		hgmapfval.s	. . . . . . . . . . . . . . 15 ⊢ ∙ = ( ·𝑠 ‘𝐶)
51	49, 50	eqtr4di 2814	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ∙ )
52		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → 𝑦 = 𝑦)
53	40	fveq1d 6879	. . . . . . . . . . . . . 14 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑚‘𝑣) = (𝑀‘𝑣))
54	51, 52, 53	oveq123d 7433	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑈 ∧ 𝑏 = 𝐵 ∧ 𝑚 = 𝑀) → (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(𝑚‘𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))
55	45, 54	eqeq12d 2777	. . . . . . . . . . . 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 3335	. . . . . . . . . . 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 7372	. . . . . . . . . 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 2847	. . . . . . . 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 1438	. . . . . . 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 3816	. . . . . 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 2895	. . . 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 2761	. . . 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 7226	. . 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 2816	. 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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  [wsbc 3739   ↦ cmpt 5186  ‘cfv 6531  ℩crio 7368  (class class class)co 7412  Basecbs 17367  Scalarcsca 17411   ·_𝑠 cvsca 17412  LHypclh 41009  DVecHcdvh 42103  LCDualclcd 42611  HDMapchdma 42817  HGMapchg 42908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-hgmap 42909

This theorem is used by:  hgmapval  42912  hgmapfnN  42913