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Theorem dvaset 35773
Description: The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvaset.h 𝐻 = (LHyp‘𝐾)
dvaset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvaset.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvaset.d 𝐷 = ((EDRing‘𝐾)‘𝑊)
dvaset.u 𝑈 = ((DVecA‘𝐾)‘𝑊)
Assertion
Ref Expression
dvaset ((𝐾𝑋𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑓,𝑊,𝑔,𝑠
Allowed substitution hints:   𝐷(𝑓,𝑔,𝑠)   𝑇(𝑓,𝑔,𝑠)   𝑈(𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔,𝑠)   𝐻(𝑓,𝑔,𝑠)   𝑋(𝑓,𝑔,𝑠)

Proof of Theorem dvaset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dvaset.u . 2 𝑈 = ((DVecA‘𝐾)‘𝑊)
2 dvaset.h . . . . 5 𝐻 = (LHyp‘𝐾)
32dvafset 35772 . . . 4 (𝐾𝑋 → (DVecA‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩})))
43fveq1d 6150 . . 3 (𝐾𝑋 → ((DVecA‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩}))‘𝑊))
5 fveq2 6148 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6 dvaset.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
75, 6syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
87opeq2d 4377 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), 𝑇⟩)
9 eqidd 2622 . . . . . . . 8 (𝑤 = 𝑊 → (𝑓𝑔) = (𝑓𝑔))
107, 7, 9mpt2eq123dv 6670 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔)) = (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔)))
1110opeq2d 4377 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩)
12 fveq2 6148 . . . . . . . 8 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
13 dvaset.d . . . . . . . 8 𝐷 = ((EDRing‘𝐾)‘𝑊)
1412, 13syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = 𝐷)
1514opeq2d 4377 . . . . . 6 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩ = ⟨(Scalar‘ndx), 𝐷⟩)
168, 11, 15tpeq123d 4253 . . . . 5 (𝑤 = 𝑊 → {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩})
17 fveq2 6148 . . . . . . . . 9 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
18 dvaset.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
1917, 18syl6eqr 2673 . . . . . . . 8 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
20 eqidd 2622 . . . . . . . 8 (𝑤 = 𝑊 → (𝑠𝑓) = (𝑠𝑓))
2119, 7, 20mpt2eq123dv 6670 . . . . . . 7 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓)) = (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓)))
2221opeq2d 4377 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩ = ⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩)
2322sneqd 4160 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩} = {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩})
2416, 23uneq12d 3746 . . . 4 (𝑤 = 𝑊 → ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩}) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
25 eqid 2621 . . . 4 (𝑤𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩})) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩}))
26 tpex 6910 . . . . 5 {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∈ V
27 snex 4869 . . . . 5 {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩} ∈ V
2826, 27unex 6909 . . . 4 ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}) ∈ V
2924, 25, 28fvmpt 6239 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩}))‘𝑊) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
304, 29sylan9eq 2675 . 2 ((𝐾𝑋𝑊𝐻) → ((DVecA‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
311, 30syl5eq 2667 1 ((𝐾𝑋𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cun 3553  {csn 4148  {ctp 4152  cop 4154  cmpt 4673  ccom 5078  cfv 5847  cmpt2 6606  ndxcnx 15778  Basecbs 15781  +gcplusg 15862  Scalarcsca 15865   ·𝑠 cvsca 15866  LHypclh 34750  LTrncltrn 34867  TEndoctendo 35520  EDRingcedring 35521  DVecAcdveca 35770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-dveca 35771
This theorem is referenced by:  dvasca  35774  dvavbase  35781  dvafvadd  35782  dvafvsca  35784  dvaabl  35793
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