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Theorem dvaset 39468
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 39467 . . . 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 6844 . . 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 6842 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6 dvaset.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
75, 6eqtr4di 2794 . . . . . . 7 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
87opeq2d 4837 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), 𝑇⟩)
9 eqidd 2737 . . . . . . . 8 (𝑤 = 𝑊 → (𝑓𝑔) = (𝑓𝑔))
107, 7, 9mpoeq123dv 7432 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔)) = (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔)))
1110opeq2d 4837 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩)
12 fveq2 6842 . . . . . . . 8 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
13 dvaset.d . . . . . . . 8 𝐷 = ((EDRing‘𝐾)‘𝑊)
1412, 13eqtr4di 2794 . . . . . . 7 (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = 𝐷)
1514opeq2d 4837 . . . . . 6 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩ = ⟨(Scalar‘ndx), 𝐷⟩)
168, 11, 15tpeq123d 4709 . . . . 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 6842 . . . . . . . . 9 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
18 dvaset.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
1917, 18eqtr4di 2794 . . . . . . . 8 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
20 eqidd 2737 . . . . . . . 8 (𝑤 = 𝑊 → (𝑠𝑓) = (𝑠𝑓))
2119, 7, 20mpoeq123dv 7432 . . . . . . 7 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓)) = (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓)))
2221opeq2d 4837 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩ = ⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩)
2322sneqd 4598 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩} = {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩})
2416, 23uneq12d 4124 . . . 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 2736 . . . 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 7681 . . . . 5 {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∈ V
27 snex 5388 . . . . 5 {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩} ∈ V
2826, 27unex 7680 . . . 4 ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}) ∈ V
2924, 25, 28fvmpt 6948 . . 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 2796 . 2 ((𝐾𝑋𝑊𝐻) → ((DVecA‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
311, 30eqtrid 2788 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 396   = wceq 1541  wcel 2106  cun 3908  {csn 4586  {ctp 4590  cop 4592  cmpt 5188  ccom 5637  cfv 6496  cmpo 7359  ndxcnx 17065  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136   ·𝑠 cvsca 17137  LHypclh 38447  LTrncltrn 38564  TEndoctendo 39215  EDRingcedring 39216  DVecAcdveca 39465
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-oprab 7361  df-mpo 7362  df-dveca 39466
This theorem is referenced by:  dvasca  39469  dvavbase  39476  dvafvadd  39477  dvafvsca  39479  dvaabl  39487
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