Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvaset Structured version   Visualization version   GIF version

Theorem dvaset 35311
 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 35310 . . . 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 6105 . . 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 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6 dvaset.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
75, 6syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
87opeq2d 4347 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), 𝑇⟩)
9 eqidd 2611 . . . . . . . 8 (𝑤 = 𝑊 → (𝑓𝑔) = (𝑓𝑔))
107, 7, 9mpt2eq123dv 6615 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔)) = (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔)))
1110opeq2d 4347 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩)
12 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
13 dvaset.d . . . . . . . 8 𝐷 = ((EDRing‘𝐾)‘𝑊)
1412, 13syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = 𝐷)
1514opeq2d 4347 . . . . . 6 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩ = ⟨(Scalar‘ndx), 𝐷⟩)
168, 11, 15tpeq123d 4227 . . . . 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 6103 . . . . . . . . 9 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
18 dvaset.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
1917, 18syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
20 eqidd 2611 . . . . . . . 8 (𝑤 = 𝑊 → (𝑠𝑓) = (𝑠𝑓))
2119, 7, 20mpt2eq123dv 6615 . . . . . . 7 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓)) = (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓)))
2221opeq2d 4347 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩ = ⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩)
2322sneqd 4137 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩} = {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩})
2416, 23uneq12d 3730 . . . 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 2610 . . . 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 6855 . . . . 5 {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∈ V
27 snex 4835 . . . . 5 {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩} ∈ V
2826, 27unex 6854 . . . 4 ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}) ∈ V
2924, 25, 28fvmpt 6191 . . 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 2664 . 2 ((𝐾𝑋𝑊𝐻) → ((DVecA‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
311, 30syl5eq 2656 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 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {csn 4125  {ctp 4129  ⟨cop 4131   ↦ cmpt 4643   ∘ ccom 5042  ‘cfv 5804   ↦ cmpt2 6551  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058  EDRingcedring 35059  DVecAcdveca 35308 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-dveca 35309 This theorem is referenced by:  dvasca  35312  dvavbase  35319  dvafvadd  35320  dvafvsca  35322  dvaabl  35331
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