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Theorem ldualset 33228
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 us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us 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 3179 . 2 (𝑊𝑋𝑊 ∈ V)
3 ldualset.d . . 3 𝐷 = (LDual‘𝑊)
4 fveq2 6083 . . . . . . . 8 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5 ldualset.f . . . . . . . 8 𝐹 = (LFnl‘𝑊)
64, 5syl6eqr 2656 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
76opeq2d 4336 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8 fveq2 6083 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 ldualset.r . . . . . . . . . . . . 13 𝑅 = (Scalar‘𝑊)
108, 9syl6eqr 2656 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
1110fveq2d 6087 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g𝑅))
12 ldualset.a . . . . . . . . . . 11 + = (+g𝑅)
1311, 12syl6eqr 2656 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14 ofeq 6769 . . . . . . . . . 10 ((+g‘(Scalar‘𝑤)) = + → ∘𝑓 (+g‘(Scalar‘𝑤)) = ∘𝑓 + )
1513, 14syl 17 . . . . . . . . 9 (𝑤 = 𝑊 → ∘𝑓 (+g‘(Scalar‘𝑤)) = ∘𝑓 + )
166sqxpeqd 5050 . . . . . . . . 9 (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
1715, 16reseq12d 5300 . . . . . . . 8 (𝑤 = 𝑊 → ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘𝑓 + ↾ (𝐹 × 𝐹)))
18 ldualset.p . . . . . . . 8 = ( ∘𝑓 + ↾ (𝐹 × 𝐹))
1917, 18syl6eqr 2656 . . . . . . 7 (𝑤 = 𝑊 → ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = )
2019opeq2d 4336 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ⟩)
2110fveq2d 6087 . . . . . . . 8 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr𝑅))
22 ldualset.o . . . . . . . 8 𝑂 = (oppr𝑅)
2321, 22syl6eqr 2656 . . . . . . 7 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
2423opeq2d 4336 . . . . . 6 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
257, 20, 24tpeq123d 4221 . . . . 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⟩})
2610fveq2d 6087 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
27 ldualset.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
2826, 27syl6eqr 2656 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
2910fveq2d 6087 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r𝑅))
30 ldualset.t . . . . . . . . . . . 12 · = (.r𝑅)
3129, 30syl6eqr 2656 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
32 ofeq 6769 . . . . . . . . . . 11 ((.r‘(Scalar‘𝑤)) = · → ∘𝑓 (.r‘(Scalar‘𝑤)) = ∘𝑓 · )
3331, 32syl 17 . . . . . . . . . 10 (𝑤 = 𝑊 → ∘𝑓 (.r‘(Scalar‘𝑤)) = ∘𝑓 · )
34 eqidd 2605 . . . . . . . . . 10 (𝑤 = 𝑊𝑓 = 𝑓)
35 fveq2 6083 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
36 ldualset.v . . . . . . . . . . . 12 𝑉 = (Base‘𝑊)
3735, 36syl6eqr 2656 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
3837xpeq1d 5047 . . . . . . . . . 10 (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
3933, 34, 38oveq123d 6543 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓𝑓 · (𝑉 × {𝑘})))
4028, 6, 39mpt2eq123dv 6588 . . . . . . . 8 (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓𝑓 · (𝑉 × {𝑘}))))
41 ldualset.s . . . . . . . 8 = (𝑘𝐾, 𝑓𝐹 ↦ (𝑓𝑓 · (𝑉 × {𝑘})))
4240, 41syl6eqr 2656 . . . . . . 7 (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = )
4342opeq2d 4336 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
4443sneqd 4131 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨( ·𝑠 ‘ndx), ⟩})
4525, 44uneq12d 3724 . . . 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), ⟩}))
46 df-ldual 33227 . . . 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})))⟩}))
47 tpex 6827 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
48 snex 4825 . . . . 5 {⟨( ·𝑠 ‘ndx), ⟩} ∈ V
4947, 48unex 6826 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}) ∈ V
5045, 46, 49fvmpt 6171 . . 3 (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
513, 50syl5eq 2650 . 2 (𝑊 ∈ V → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
521, 2, 513syl 18 1 (𝜑𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3167  cun 3532  {csn 4119  {ctp 4123  cop 4125   × cxp 5021  cres 5025  cfv 5785  (class class class)co 6522  cmpt2 6524  𝑓 cof 6765  ndxcnx 15633  Basecbs 15636  +gcplusg 15709  .rcmulr 15710  Scalarcsca 15712   ·𝑠 cvsca 15713  opprcoppr 18386  LFnlclfn 33160  LDualcld 33226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-res 5035  df-iota 5749  df-fun 5787  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-ldual 33227
This theorem is referenced by:  ldualvbase  33229  ldualfvadd  33231  ldualsca  33235  ldualfvs  33239
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