Theorem hlhilset 40397

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 3463	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ V)
4	3	adantr 481	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ V)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
6		hlhilset.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
7	6	fvexi 6856	. . . . 5 ⊢ 𝐻 ∈ V
8	7	mptex 7173	. . . 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 2907	. . . . 5 ⊢ Ⅎ𝑘𝐾
10		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑘𝐻
11		nfcsb1v 3880	. . . . . 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 5212	. . . . 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 6842	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
14	13, 6	eqtr4di 2794	. . . . . 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 5196	. . . . 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 40396	. . . . 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 6969	. . . 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 586	. . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑤 = 𝑊) → 𝐾 ∈ V)
21		fvexd 6857	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) ∈ V)
22		fvexd 6857	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) ∈ V)
23		id 22	. . . . . . . . . 10 ⊢ (𝑣 = (Base‘𝑢) → 𝑣 = (Base‘𝑢))
24		id 22	. . . . . . . . . . . . 13 ⊢ (𝑢 = ((DVecH‘𝑘)‘𝑤) → 𝑢 = ((DVecH‘𝑘)‘𝑤))
25		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
26	25	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (DVecH‘𝑘) = (DVecH‘𝐾))
27		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → 𝑤 = 𝑊)
28	26, 27	fveq12d 6849	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
29		hlhilset.u	. . . . . . . . . . . . . 14 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
30	28, 29	eqtr4di 2794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((DVecH‘𝑘)‘𝑤) = 𝑈)
31	24, 30	sylan9eqr 2798	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → 𝑢 = 𝑈)
32	31	fveq2d 6846	. . . . . . . . . . 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 2794	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) → (Base‘𝑢) = 𝑉)
35	23, 34	sylan9eqr 2798	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑣 = 𝑉)
36	35	opeq2d 4837	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(Base‘ndx), 𝑣⟩ = ⟨(Base‘ndx), 𝑉⟩)
37	31	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → 𝑢 = 𝑈)
38	37	fveq2d 6846	. . . . . . . . . 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 2794	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (+g‘𝑢) = + )
41	40	opeq2d 4837	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨(+g‘ndx), (+g‘𝑢)⟩ = ⟨(+g‘ndx), + ⟩)
42	25	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (EDRing‘𝑘) = (EDRing‘𝐾))
43	42, 27	fveq12d 6849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
44		hlhilset.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)
45	43, 44	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((EDRing‘𝑘)‘𝑤) = 𝐸)
46	25	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HGMap‘𝑘) = (HGMap‘𝐾))
47	46, 27	fveq12d 6849	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = ((HGMap‘𝐾)‘𝑊))
48		hlhilset.g	. . . . . . . . . . . . . 14 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
49	47, 48	eqtr4di 2794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HGMap‘𝑘)‘𝑤) = 𝐺)
50	49	opeq2d 4837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩ = ⟨(*𝑟‘ndx), 𝐺⟩)
51	45, 50	oveq12d 7375	. . . . . . . . . . 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 2794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩) = 𝑅)
54	53	opeq2d 4837	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
55	54	ad2antrr 724	. . . . . . . 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 4709	. . . . . . 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 6846	. . . . . . . . . 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 2794	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ( ·𝑠 ‘𝑢) = · )
60	59	opeq2d 4837	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩ = ⟨( ·𝑠 ‘ndx), · ⟩)
61	25	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → (HDMap‘𝑘) = (HDMap‘𝐾))
62	61, 27	fveq12d 6849	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = ((HDMap‘𝐾)‘𝑊))
63		hlhilset.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
64	62, 63	eqtr4di 2794	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
65	64	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → ((HDMap‘𝑘)‘𝑤) = 𝑆)
66	65	fveq1d 6844	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑤 = 𝑊) ∧ 𝑘 = 𝐾) ∧ 𝑢 = ((DVecH‘𝑘)‘𝑤)) ∧ 𝑣 = (Base‘𝑢)) → (((HDMap‘𝑘)‘𝑤)‘𝑦) = (𝑆‘𝑦))
67	66	fveq1d 6844	. . . . . . . . . . 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 7432	. . . . . . . . . 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 2794	. . . . . . . . 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 4837	. . . . . . . 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 4702	. . . . . . 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 4124	. . . . . 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 3893	. . . . 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 3893	. . . 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 3893	. . 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 496	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
78		tpex 7681	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∈ V
79		prex 5389	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩} ∈ V
80	78, 79	unex 7680	. . . 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 6955	. 2 ⊢ (𝜑 → ((HLHil‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
83	1, 82	eqtrid 2788	1 ⊢ (𝜑 → 𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  ⦋csb 3855   ∪ cun 3908  {cpr 4588  {ctp 4590  ⟨cop 4592   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359   sSet csts 17035  ndxcnx 17065  Basecbs 17083  +_gcplusg 17133  *_𝑟cstv 17135  Scalarcsca 17136   ·_𝑠 cvsca 17137  ·_𝑖cip 17138  HLchlt 37812  LHypclh 38447  EDRingcedring 39216  DVecHcdvh 39541  HDMapchdma 40255  HGMapchg 40346  HLHilchlh 40395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-hlhil 40396

This theorem is referenced by:  hlhilsca  40398  hlhilbase  40399  hlhilplus  40400  hlhilvsca  40414  hlhilip  40415