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

Theorem ldualset 37995
Description: Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
Hypotheses
Ref Expression
ldualset.v 𝑉 = (Base‘𝑊)
ldualset.a + = (+g𝑅)
ldualset.p = ( ∘f + ↾ (𝐹 × 𝐹))
ldualset.f 𝐹 = (LFnl‘𝑊)
ldualset.d 𝐷 = (LDual‘𝑊)
ldualset.r 𝑅 = (Scalar‘𝑊)
ldualset.k 𝐾 = (Base‘𝑅)
ldualset.t · = (.r𝑅)
ldualset.o 𝑂 = (oppr𝑅)
ldualset.s = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓f · (𝑉 × {𝑘})))
ldualset.w (𝜑𝑊𝑋)
Assertion
Ref Expression
ldualset (𝜑𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
Distinct variable group:   𝑓,𝑘,𝑊
Allowed substitution hints:   𝜑(𝑓,𝑘)   𝐷(𝑓,𝑘)   + (𝑓,𝑘)   (𝑓,𝑘)   𝑅(𝑓,𝑘)   (𝑓,𝑘)   · (𝑓,𝑘)   𝐹(𝑓,𝑘)   𝐾(𝑓,𝑘)   𝑂(𝑓,𝑘)   𝑉(𝑓,𝑘)   𝑋(𝑓,𝑘)

Proof of Theorem ldualset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ldualset.w . 2 (𝜑𝑊𝑋)
2 elex 3493 . 2 (𝑊𝑋𝑊 ∈ V)
3 ldualset.d . . 3 𝐷 = (LDual‘𝑊)
4 fveq2 6892 . . . . . . . 8 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5 ldualset.f . . . . . . . 8 𝐹 = (LFnl‘𝑊)
64, 5eqtr4di 2791 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
76opeq2d 4881 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8 fveq2 6892 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 ldualset.r . . . . . . . . . . . . 13 𝑅 = (Scalar‘𝑊)
108, 9eqtr4di 2791 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
1110fveq2d 6896 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g𝑅))
12 ldualset.a . . . . . . . . . . 11 + = (+g𝑅)
1311, 12eqtr4di 2791 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
1413ofeqd 7672 . . . . . . . . 9 (𝑤 = 𝑊 → ∘f (+g‘(Scalar‘𝑤)) = ∘f + )
156sqxpeqd 5709 . . . . . . . . 9 (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
1614, 15reseq12d 5983 . . . . . . . 8 (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘f + ↾ (𝐹 × 𝐹)))
17 ldualset.p . . . . . . . 8 = ( ∘f + ↾ (𝐹 × 𝐹))
1816, 17eqtr4di 2791 . . . . . . 7 (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = )
1918opeq2d 4881 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ⟩)
2010fveq2d 6896 . . . . . . . 8 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr𝑅))
21 ldualset.o . . . . . . . 8 𝑂 = (oppr𝑅)
2220, 21eqtr4di 2791 . . . . . . 7 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
2322opeq2d 4881 . . . . . 6 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
247, 19, 23tpeq123d 4753 . . . . 5 (𝑤 = 𝑊 → {⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩})
2510fveq2d 6896 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
26 ldualset.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
2725, 26eqtr4di 2791 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
2810fveq2d 6896 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r𝑅))
29 ldualset.t . . . . . . . . . . . 12 · = (.r𝑅)
3028, 29eqtr4di 2791 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
3130ofeqd 7672 . . . . . . . . . 10 (𝑤 = 𝑊 → ∘f (.r‘(Scalar‘𝑤)) = ∘f · )
32 eqidd 2734 . . . . . . . . . 10 (𝑤 = 𝑊𝑓 = 𝑓)
33 fveq2 6892 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
34 ldualset.v . . . . . . . . . . . 12 𝑉 = (Base‘𝑊)
3533, 34eqtr4di 2791 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
3635xpeq1d 5706 . . . . . . . . . 10 (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
3731, 32, 36oveq123d 7430 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓f · (𝑉 × {𝑘})))
3827, 6, 37mpoeq123dv 7484 . . . . . . . 8 (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓f · (𝑉 × {𝑘}))))
39 ldualset.s . . . . . . . 8 = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓f · (𝑉 × {𝑘})))
4038, 39eqtr4di 2791 . . . . . . 7 (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = )
4140opeq2d 4881 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
4241sneqd 4641 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨( ·𝑠 ‘ndx), ⟩})
4324, 42uneq12d 4165 . . . 4 (𝑤 = 𝑊 → ({⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
44 df-ldual 37994 . . . 4 LDual = (𝑤 ∈ V ↦ ({⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩}))
45 tpex 7734 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
46 snex 5432 . . . . 5 {⟨( ·𝑠 ‘ndx), ⟩} ∈ V
4745, 46unex 7733 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}) ∈ V
4843, 44, 47fvmpt 6999 . . 3 (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
493, 48eqtrid 2785 . 2 (𝑊 ∈ V → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
501, 2, 493syl 18 1 (𝜑𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cun 3947  {csn 4629  {ctp 4633  cop 4635   × cxp 5675  cres 5679  cfv 6544  (class class class)co 7409  cmpo 7411  f cof 7668  ndxcnx 17126  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  opprcoppr 20149  LFnlclfn 37927  LDualcld 37993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-ldual 37994
This theorem is referenced by:  ldualvbase  37996  ldualfvadd  37998  ldualsca  38002  ldualfvs  38006
  Copyright terms: Public domain W3C validator