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Theorem ldualset 35081
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 3365 . 2 (𝑊𝑋𝑊 ∈ V)
3 ldualset.d . . 3 𝐷 = (LDual‘𝑊)
4 fveq2 6375 . . . . . . . 8 (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5 ldualset.f . . . . . . . 8 𝐹 = (LFnl‘𝑊)
64, 5syl6eqr 2817 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
76opeq2d 4566 . . . . . 6 (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8 fveq2 6375 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 ldualset.r . . . . . . . . . . . . 13 𝑅 = (Scalar‘𝑊)
108, 9syl6eqr 2817 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
1110fveq2d 6379 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g𝑅))
12 ldualset.a . . . . . . . . . . 11 + = (+g𝑅)
1311, 12syl6eqr 2817 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14 ofeq 7097 . . . . . . . . . 10 ((+g‘(Scalar‘𝑤)) = + → ∘𝑓 (+g‘(Scalar‘𝑤)) = ∘𝑓 + )
1513, 14syl 17 . . . . . . . . 9 (𝑤 = 𝑊 → ∘𝑓 (+g‘(Scalar‘𝑤)) = ∘𝑓 + )
166sqxpeqd 5309 . . . . . . . . 9 (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
1715, 16reseq12d 5566 . . . . . . . 8 (𝑤 = 𝑊 → ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘𝑓 + ↾ (𝐹 × 𝐹)))
18 ldualset.p . . . . . . . 8 = ( ∘𝑓 + ↾ (𝐹 × 𝐹))
1917, 18syl6eqr 2817 . . . . . . 7 (𝑤 = 𝑊 → ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = )
2019opeq2d 4566 . . . . . 6 (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ⟩)
2110fveq2d 6379 . . . . . . . 8 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr𝑅))
22 ldualset.o . . . . . . . 8 𝑂 = (oppr𝑅)
2321, 22syl6eqr 2817 . . . . . . 7 (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
2423opeq2d 4566 . . . . . 6 (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
257, 20, 24tpeq123d 4438 . . . . 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 6379 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
27 ldualset.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
2826, 27syl6eqr 2817 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
2910fveq2d 6379 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r𝑅))
30 ldualset.t . . . . . . . . . . . 12 · = (.r𝑅)
3129, 30syl6eqr 2817 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
32 ofeq 7097 . . . . . . . . . . 11 ((.r‘(Scalar‘𝑤)) = · → ∘𝑓 (.r‘(Scalar‘𝑤)) = ∘𝑓 · )
3331, 32syl 17 . . . . . . . . . 10 (𝑤 = 𝑊 → ∘𝑓 (.r‘(Scalar‘𝑤)) = ∘𝑓 · )
34 eqidd 2766 . . . . . . . . . 10 (𝑤 = 𝑊𝑓 = 𝑓)
35 fveq2 6375 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
36 ldualset.v . . . . . . . . . . . 12 𝑉 = (Base‘𝑊)
3735, 36syl6eqr 2817 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
3837xpeq1d 5306 . . . . . . . . . 10 (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
3933, 34, 38oveq123d 6863 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓𝑓 · (𝑉 × {𝑘})))
4028, 6, 39mpt2eq123dv 6915 . . . . . . . 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 2817 . . . . . . 7 (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = )
4342opeq2d 4566 . . . . . 6 (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
4443sneqd 4346 . . . . 5 (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨( ·𝑠 ‘ndx), ⟩})
4525, 44uneq12d 3930 . . . 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 35080 . . . 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 7155 . . . . 5 {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
48 snex 5064 . . . . 5 {⟨( ·𝑠 ‘ndx), ⟩} ∈ V
4947, 48unex 7154 . . . 4 ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}) ∈ V
5045, 46, 49fvmpt 6471 . . 3 (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ⟩}))
513, 50syl5eq 2811 . 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 1652  wcel 2155  Vcvv 3350  cun 3730  {csn 4334  {ctp 4338  cop 4340   × cxp 5275  cres 5279  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑓 cof 7093  ndxcnx 16127  Basecbs 16130  +gcplusg 16214  .rcmulr 16215  Scalarcsca 16217   ·𝑠 cvsca 16218  opprcoppr 18889  LFnlclfn 35013  LDualcld 35079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ldual 35080
This theorem is referenced by:  ldualvbase  35082  ldualfvadd  35084  ldualsca  35088  ldualfvs  35092
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