dvhfset - Mathbox for Norm Megill

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Theorem dvhfset 42105

Description: The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

**Hypothesis**
Ref	Expression
dvhset.h	⊢ 𝐻 = (LHyp‘𝐾)

**Assertion**
Ref	Expression
dvhfset	⊢ (𝐾 ∈ 𝑉 → (DVecH‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))

Distinct variable groups: 𝑓,𝑔,𝑤,𝐻 𝑓,ℎ,𝑠,𝐾,𝑔,𝑤 Allowed substitution hints: 𝐻(ℎ, 𝑠) 𝑉(𝑤, 𝑓, 𝑔, ℎ, 𝑠)

Proof of Theorem dvhfset Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3472	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6877	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dvhset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2814	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6877	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6879	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7		fveq2 6877	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
8	7	fveq1d 6879	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
9	6, 8	xpeq12d 5682	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) = (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)))
10	9	opeq2d 4840	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩ = ⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩)
11	6	mpteq1d 5195	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ))))
12	11	opeq2d 4840	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩ = ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)
13	9, 9, 12	mpoeq123dv 7487	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩) = (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩))
14	13	opeq2d 4840	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩ = ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩)
15		fveq2 6877	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (EDRing‘𝑘) = (EDRing‘𝐾))
16	15	fveq1d 6879	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑤))
17	16	opeq2d 4840	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩ = ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩)
18	10, 14, 17	tpeq123d 4709	. . . . 5 ⊢ (𝑘 = 𝐾 → {⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} = {⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩})
19		eqidd 2762	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩ = ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)
20	8, 9, 19	mpoeq123dv 7487	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩))
21	20	opeq2d 4840	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩ = ⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩)
22	21	sneqd 4596	. . . . 5 ⊢ (𝑘 = 𝐾 → {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩} = {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})
23	18, 22	uneq12d 4116	. . . 4 ⊢ (𝑘 = 𝐾 → ({⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩}))
24	4, 23	mpteq12dv 5192	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))
25		df-dvech 42104	. . 3 ⊢ DVecH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))
26	24, 25, 3	mptfvmpt 7226	. 2 ⊢ (𝐾 ∈ V → (DVecH‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))
27	1, 26	syl 18	1 ⊢ (𝐾 ∈ 𝑉 → (DVecH‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+_g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2^nd ‘𝑓)‘ℎ) ∘ ((2^nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·_𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)⟩})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ∪ cun 3897 {csn 4584 {ctp 4588 ⟨cop 4590 ↦ cmpt 5186 × cxp 5649 ∘ ccom 5655 ‘cfv 6531 ∈ cmpo 7414 1^st c1st 7988 2^nd c2nd 7989 ndxcnx 17351 Basecbs 17367 +_gcplusg 17408 Scalarcsca 17411 ·_𝑠 cvsca 17412 LHypclh 41009 LTrncltrn 41126 TEndoctendo 41777 EDRingcedring 41778 DVecHcdvh 42103

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-tp 4589 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-oprab 7416 df-mpo 7417 df-dvech 42104

This theorem is used by: dvhset 42106

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