Theorem hlhilset 42570

Description: The final Hilbert space constructed from a Hilbert lattice 𝐾 and an arbitrary hyperplane 𝑊 in 𝐾. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)

Hypotheses

Ref	Expression
hlhilset.h	⊢ 𝐻 = (LHyp‘𝐾)
hlhilset.l	⊢ 𝐿 = ((HLHil‘𝐾)‘𝑊)
hlhilset.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hlhilset.v	⊢ 𝑉 = (Base‘𝑈)
hlhilset.p	⊢ + = (+g‘𝑈)
hlhilset.e	⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)
hlhilset.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hlhilset.r	⊢ 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)
hlhilset.t	⊢ · = ( ·𝑠 ‘𝑈)
hlhilset.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hlhilset.i	⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))
hlhilset.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
hlhilset	⊢ (𝜑 → 𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))

Distinct variable groups:   𝑥,𝑦,𝐾   𝜑,𝑥,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   + (𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   · (𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   , (𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem hlhilset

Dummy variables 𝑤 𝑘 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlhilset.l	. 2 ⊢ 𝐿 = ((HLHil‘𝐾)‘𝑊)
2		hlhilset.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		elex 3478	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ V)
4	3	adantr 485	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ V)
5	2, 4	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
6		hlhilset.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
7	6	fvexi 6885	. . . . 5 ⊢ 𝐻 ∈ V
8	7	mptex 7211	. . . 4 ⊢ (𝑤 ∈ 𝐻 ↦ ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})) ∈ V
9		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑘𝐾
10		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑘𝐻
11		nfcsb1v 3879	. . . . . 6 ⊢ Ⅎ𝑘⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})
12	10, 11	nfmpt 5203	. . . . 5 ⊢ Ⅎ𝑘(𝑤 ∈ 𝐻 ↦ ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}))
13		fveq2 6871	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
14	13, 6	eqtr4di 2818	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
15		csbeq1a 3869	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}))
16	14, 15	mpteq12dv 5192	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})) = (𝑤 ∈ 𝐻 ↦ ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
17		df-hlhil 42569	. . . . 5 ⊢ HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
18	9, 12, 16, 17	fvmptf 7001	. . . 4 ⊢ ((𝐾 ∈ V ∧ (𝑤 ∈ 𝐻 ↦ ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})) ∈ V) → (HLHil‘𝐾) = (𝑤 ∈ 𝐻 ↦ ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
19	5, 8, 18	sylancl 597	. . 3 ⊢ (𝜑 → (HLHil‘𝐾) = (𝑤 ∈ 𝐻 ↦ ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
20	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → 𝐾 ∈ V)
21		fvexd 6886	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) ∈ V)
22		fvexd 6886	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) ∈ V)
23		id 23	. . . . . . . . . 10 ⊢ (𝑣 = (Base‘𝑢) → 𝑣 = (Base‘𝑢))
24		id 23	. . . . . . . . . . . . 13 ⊢ (𝑢 = ((DVecH‘𝑘)‘𝑤) → 𝑢 = ((DVecH‘𝑘)‘𝑤))
25		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
26	25	fveq2d 6875	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (DVecH‘𝑘) = (DVecH‘𝐾))
27		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑤 = 𝑊)
28	26, 27	fveq12d 6878	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
29		hlhilset.u	. . . . . . . . . . . . . 14 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
30	28, 29	eqtr4di 2818	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = 𝑈)
31	24, 30	sylan9eqr 2822	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → 𝑢 = 𝑈)
32	31	fveq2d 6875	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = (Base‘𝑈))
33		hlhilset.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Base‘𝑈)
34	32, 33	eqtr4di 2818	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = 𝑉)
35	23, 34	sylan9eqr 2822	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
36	35	opeq2d 4841	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝑉⟩)
37	31	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
38	37	fveq2d 6875	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g‘𝑢) = (+g‘𝑈))
39		hlhilset.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑈)
40	38, 39	eqtr4di 2818	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g‘𝑢) = + )
41	40	opeq2d 4841	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(+g‘ndx), (+g‘𝑢)⟩ = ⟨(+g‘ndx), + ⟩)
42	25	fveq2d 6875	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (EDRing‘𝑘) = (EDRing‘𝐾))
43	42, 27	fveq12d 6878	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
44		hlhilset.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)
45	43, 44	eqtr4di 2818	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = 𝐸)
46	25	fveq2d 6875	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HGMap‘𝑘) = (HGMap‘𝐾))
47	46, 27	fveq12d 6878	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = ((HGMap‘𝐾)‘𝑊))
48		hlhilset.g	. . . . . . . . . . . . . 14 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
49	47, 48	eqtr4di 2818	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = 𝐺)
50	49	opeq2d 4841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩ = ⟨(*𝑟‘ndx), 𝐺⟩)
51	45, 50	oveq12d 7418	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩))
52		hlhilset.r	. . . . . . . . . . 11 ⊢ 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)
53	51, 52	eqtr4di 2818	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = 𝑅)
54	53	opeq2d 4841	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
55	54	ad2antrr 738	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
56	36, 41, 55	tpeq123d 4710	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → {⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} = {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩})
57	37	fveq2d 6875	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠 ‘𝑢) = ( ·𝑠 ‘𝑈))
58		hlhilset.t	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘𝑈)
59	57, 58	eqtr4di 2818	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠 ‘𝑢) = · )
60	59	opeq2d 4841	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
61	25	fveq2d 6875	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HDMap‘𝑘) = (HDMap‘𝐾))
62	61, 27	fveq12d 6878	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
63		hlhilset.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
64	62, 63	eqtr4di 2818	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
65	64	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
66	65	fveq1d 6873	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (((HDMap‘𝑘)‘𝑤)‘𝑦) = (𝑆‘𝑦))
67	66	fveq1d 6873	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥) = ((𝑆‘𝑦)‘𝑥))
68	35, 35, 67	mpoeq123dv 7475	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)))
69		hlhilset.i	. . . . . . . . . 10 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))
70	68, 69	eqtr4di 2818	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥)) = , )
71	70	opeq2d 4841	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩ = ⟨(·𝑖‘ndx), , ⟩)
72	60, 71	preq12d 4703	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩} = {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩})
73	56, 72	uneq12d 4125	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
74	22, 73	csbied 3891	. . . . 5 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → ⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
75	21, 74	csbied 3891	. . . 4 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
76	20, 75	csbied 3891	. . 3 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → ⦋𝐾 / 𝑘⦌⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩}) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
77	2	simprd 500	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
78		tpex 7733	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∈ V
79		prex 5400	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
80	78, 79	unex 7731	. . . 4 ⊢ ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V
81	80	a1i 11	. . 3 ⊢ (𝜑 → ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∈ V)
82	19, 76, 77, 81	fvmptd 6987	. 2 ⊢ (𝜑 → ((HLHil‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
83	1, 82	eqtrid 2812	1 ⊢ (𝜑 → 𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ⦋csb 3855   ∪ cun 3905  {cpr 4587  {ctp 4589  ⟨cop 4591   ↦ cmpt 5186  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   sSet csts 17213  ndxcnx 17243  Basecbs 17259  +_gcplusg 17300  *_𝑟cstv 17302  Scalarcsca 17303   ·_𝑠 cvsca 17304  ·_𝑖cip 17305  HLchlt 39986  LHypclh 40620  EDRingcedring 41389  DVecHcdvh 41714  HDMapchdma 42428  HGMapchg 42519  HLHilchlh 42568

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  ax-un 7722

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-tp 4590  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-ov 7403  df-oprab 7404  df-mpo 7405  df-hlhil 42569

This theorem is referenced by:  hlhilsca  42571  hlhilbase  42572  hlhilplus  42573  hlhilvsca  42583  hlhilip  42584