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Theorem dvhfset 36871
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 (𝐾𝑉 → (DVecH‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
Distinct variable groups:   𝑓,𝑔,𝑤,𝐻   𝑓,,𝑠,𝐾,𝑔,𝑤
Allowed substitution hints:   𝐻(,𝑠)   𝑉(𝑤,𝑓,𝑔,,𝑠)

Proof of Theorem dvhfset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3352 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6352 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dvhset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2812 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6352 . . . . . . . . 9 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
65fveq1d 6354 . . . . . . . 8 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7 fveq2 6352 . . . . . . . . 9 (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
87fveq1d 6354 . . . . . . . 8 (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
96, 8xpeq12d 5297 . . . . . . 7 (𝑘 = 𝐾 → (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) = (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)))
109opeq2d 4560 . . . . . 6 (𝑘 = 𝐾 → ⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩ = ⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩)
116mpteq1d 4890 . . . . . . . . 9 (𝑘 = 𝐾 → ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))) = ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))))
1211opeq2d 4560 . . . . . . . 8 (𝑘 = 𝐾 → ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩ = ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)
139, 9, 12mpt2eq123dv 6882 . . . . . . 7 (𝑘 = 𝐾 → (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩) = (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩))
1413opeq2d 4560 . . . . . 6 (𝑘 = 𝐾 → ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩ = ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩)
15 fveq2 6352 . . . . . . . 8 (𝑘 = 𝐾 → (EDRing‘𝑘) = (EDRing‘𝐾))
1615fveq1d 6354 . . . . . . 7 (𝑘 = 𝐾 → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑤))
1716opeq2d 4560 . . . . . 6 (𝑘 = 𝐾 → ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩ = ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩)
1810, 14, 17tpeq123d 4427 . . . . 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)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(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)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩})
19 eqidd 2761 . . . . . . . 8 (𝑘 = 𝐾 → ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩ = ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
208, 9, 19mpt2eq123dv 6882 . . . . . . 7 (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
2120opeq2d 4560 . . . . . 6 (𝑘 = 𝐾 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩ = ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩)
2221sneqd 4333 . . . . 5 (𝑘 = 𝐾 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩} = {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})
2318, 22uneq12d 3911 . . . 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)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
244, 23mpteq12dv 4885 . . 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)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
25 df-dvech 36870 . . 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)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
26 fvex 6362 . . . . 5 (LHyp‘𝐾) ∈ V
273, 26eqeltri 2835 . . . 4 𝐻 ∈ V
2827mptex 6650 . . 3 (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})) ∈ V
2924, 25, 28fvmpt 6444 . 2 (𝐾 ∈ V → (DVecH‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
301, 29syl 17 1 (𝐾𝑉 → (DVecH‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  {csn 4321  {ctp 4325  cop 4327  cmpt 4881   × cxp 5264  ccom 5270  cfv 6049  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  ndxcnx 16056  Basecbs 16059  +gcplusg 16143  Scalarcsca 16146   ·𝑠 cvsca 16147  LHypclh 35773  LTrncltrn 35890  TEndoctendo 36542  EDRingcedring 36543  DVecHcdvh 36869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-oprab 6817  df-mpt2 6818  df-dvech 36870
This theorem is referenced by:  dvhset  36872
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