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Theorem ocvfval 19829
Description: The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
ocvfval.v 𝑉 = (Base‘𝑊)
ocvfval.i , = (·𝑖𝑊)
ocvfval.f 𝐹 = (Scalar‘𝑊)
ocvfval.z 0 = (0g𝐹)
ocvfval.o = (ocv‘𝑊)
Assertion
Ref Expression
ocvfval (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Distinct variable groups:   𝑥,𝑠,𝑦, 0   𝑉,𝑠,𝑥,𝑦   𝑊,𝑠,𝑥,𝑦   , ,𝑠,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑠)   (𝑥,𝑦,𝑠)   𝑋(𝑥,𝑦,𝑠)

Proof of Theorem ocvfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ocvfval.o . 2 = (ocv‘𝑊)
2 elex 3185 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6103 . . . . . . 7 ( = 𝑊 → (Base‘) = (Base‘𝑊))
4 ocvfval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2662 . . . . . 6 ( = 𝑊 → (Base‘) = 𝑉)
65pweqd 4113 . . . . 5 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
7 fveq2 6103 . . . . . . . . . 10 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
8 ocvfval.i . . . . . . . . . 10 , = (·𝑖𝑊)
97, 8syl6eqr 2662 . . . . . . . . 9 ( = 𝑊 → (·𝑖) = , )
109oveqd 6566 . . . . . . . 8 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
11 fveq2 6103 . . . . . . . . . . 11 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
12 ocvfval.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1311, 12syl6eqr 2662 . . . . . . . . . 10 ( = 𝑊 → (Scalar‘) = 𝐹)
1413fveq2d 6107 . . . . . . . . 9 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
15 ocvfval.z . . . . . . . . 9 0 = (0g𝐹)
1614, 15syl6eqr 2662 . . . . . . . 8 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
1710, 16eqeq12d 2625 . . . . . . 7 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ (𝑥 , 𝑦) = 0 ))
1817ralbidv 2969 . . . . . 6 ( = 𝑊 → (∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 ))
195, 18rabeqbidv 3168 . . . . 5 ( = 𝑊 → {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))} = {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
206, 19mpteq12dv 4663 . . . 4 ( = 𝑊 → (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
21 df-ocv 19826 . . . 4 ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))
22 eqid 2610 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
23 ssrab2 3650 . . . . . . . 8 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
24 fvex 6113 . . . . . . . . . 10 (Base‘𝑊) ∈ V
254, 24eqeltri 2684 . . . . . . . . 9 𝑉 ∈ V
2625elpw2 4755 . . . . . . . 8 ({𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉 ↔ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉)
2723, 26mpbir 220 . . . . . . 7 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉
2827a1i 11 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉)
2922, 28fmpti 6291 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
3025pwex 4774 . . . . 5 𝒫 𝑉 ∈ V
31 fex2 7014 . . . . 5 (((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉 ∧ 𝒫 𝑉 ∈ V ∧ 𝒫 𝑉 ∈ V) → (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V)
3229, 30, 30, 31mp3an 1416 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
3320, 21, 32fvmpt 6191 . . 3 (𝑊 ∈ V → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
342, 33syl 17 . 2 (𝑊𝑋 → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
351, 34syl5eq 2656 1 (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  wss 3540  𝒫 cpw 4108  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771  ·𝑖cip 15773  0gc0g 15923  ocvcocv 19823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-ocv 19826
This theorem is referenced by:  ocvval  19830  elocv  19831
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