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Theorem ldualset 39107
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 3499 . 2 (𝑊𝑋𝑊 ∈ V)
3 ldualset.d . . 3 𝐷 = (LDual‘𝑊)
4 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5 ldualset.f . . . . . . . 8 𝐹 = (LFnl‘𝑊)
64, 5eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
76opeq2d 4885 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8 fveq2 6907 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 ldualset.r . . . . . . . . . . . . 13 𝑅 = (Scalar‘𝑊)
108, 9eqtr4di 2793 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
1110fveq2d 6911 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g𝑅))
12 ldualset.a . . . . . . . . . . 11 + = (+g𝑅)
1311, 12eqtr4di 2793 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
1413ofeqd 7699 . . . . . . . . 9 (𝑤 = 𝑊 → ∘f (+g‘(Scalar‘𝑤)) = ∘f + )
156sqxpeqd 5721 . . . . . . . . 9 (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
1614, 15reseq12d 6001 . . . . . . . 8 (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘f + ↾ (𝐹 × 𝐹)))
17 ldualset.p . . . . . . . 8 = ( ∘f + ↾ (𝐹 × 𝐹))
1816, 17eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = )
1918opeq2d 4885 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ⟩)
2010fveq2d 6911 . . . . . . . 8 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr𝑅))
21 ldualset.o . . . . . . . 8 𝑂 = (oppr𝑅)
2220, 21eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
2322opeq2d 4885 . . . . . 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 6911 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
26 ldualset.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
2725, 26eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
2810fveq2d 6911 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r𝑅))
29 ldualset.t . . . . . . . . . . . 12 · = (.r𝑅)
3028, 29eqtr4di 2793 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
3130ofeqd 7699 . . . . . . . . . 10 (𝑤 = 𝑊 → ∘f (.r‘(Scalar‘𝑤)) = ∘f · )
32 eqidd 2736 . . . . . . . . . 10 (𝑤 = 𝑊𝑓 = 𝑓)
33 fveq2 6907 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
34 ldualset.v . . . . . . . . . . . 12 𝑉 = (Base‘𝑊)
3533, 34eqtr4di 2793 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
3635xpeq1d 5718 . . . . . . . . . 10 (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
3731, 32, 36oveq123d 7452 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓f · (𝑉 × {𝑘})))
3827, 6, 37mpoeq123dv 7508 . . . . . . . 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 2793 . . . . . . 7 (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = )
4140opeq2d 4885 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
4241sneqd 4643 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨( ·𝑠 ‘ndx), ⟩})
4324, 42uneq12d 4179 . . . 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 39106 . . . 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 7765 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
46 snex 5442 . . . . 5 {⟨( ·𝑠 ‘ndx), ⟩} ∈ V
4745, 46unex 7763 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}) ∈ V
4843, 44, 47fvmpt 7016 . . 3 (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
493, 48eqtrid 2787 . 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 1537  wcel 2106  Vcvv 3478  cun 3961  {csn 4631  {ctp 4635  cop 4637   × cxp 5687  cres 5691  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695  ndxcnx 17227  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  opprcoppr 20350  LFnlclfn 39039  LDualcld 39105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ldual 39106
This theorem is referenced by:  ldualvbase  39108  ldualfvadd  39110  ldualsca  39114  ldualfvs  39118
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