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Theorem ocvval 21703
Description: Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (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
ocvval (𝑆𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
Distinct variable groups:   𝑥,𝑦, 0   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥, , ,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem ocvval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ocvfval.v . . . 4 𝑉 = (Base‘𝑊)
21fvexi 6921 . . 3 𝑉 ∈ V
32elpw2 5340 . 2 (𝑆 ∈ 𝒫 𝑉𝑆𝑉)
4 ocvfval.i . . . . . 6 , = (·𝑖𝑊)
5 ocvfval.f . . . . . 6 𝐹 = (Scalar‘𝑊)
6 ocvfval.z . . . . . 6 0 = (0g𝐹)
7 ocvfval.o . . . . . 6 = (ocv‘𝑊)
81, 4, 5, 6, 7ocvfval 21702 . . . . 5 (𝑊 ∈ V → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
98fveq1d 6909 . . . 4 (𝑊 ∈ V → ( 𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10 raleq 3321 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 ))
1110rabbidv 3441 . . . . 5 (𝑠 = 𝑆 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
12 eqid 2735 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
132rabex 5345 . . . . 5 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ∈ V
1411, 12, 13fvmpt 7016 . . . 4 (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
159, 14sylan9eq 2795 . . 3 ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
16 0fv 6951 . . . . 5 (∅‘𝑆) = ∅
17 fvprc 6899 . . . . . . 7 𝑊 ∈ V → (ocv‘𝑊) = ∅)
187, 17eqtrid 2787 . . . . . 6 𝑊 ∈ V → = ∅)
1918fveq1d 6909 . . . . 5 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
20 ssrab2 4090 . . . . . 6 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21 fvprc 6899 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
221, 21eqtrid 2787 . . . . . 6 𝑊 ∈ V → 𝑉 = ∅)
23 sseq0 4409 . . . . . 6 (({𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉𝑉 = ∅) → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2420, 22, 23sylancr 587 . . . . 5 𝑊 ∈ V → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2516, 19, 243eqtr4a 2801 . . . 4 𝑊 ∈ V → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2625adantr 480 . . 3 ((¬ 𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2715, 26pm2.61ian 812 . 2 (𝑆 ∈ 𝒫 𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
283, 27sylbir 235 1 (𝑆𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  ocvcocv 21696
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-pow 5371  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-ne 2939  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-pw 4607  df-sn 4632  df-pr 4634  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-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-ocv 21699
This theorem is referenced by:  elocv  21704  ocv0  21713  csscld  25297  hlhilocv  41944
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