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Theorem ocvval 20227
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)
2 fvex 6342 . . . 4 (Base‘𝑊) ∈ V
31, 2eqeltri 2845 . . 3 𝑉 ∈ V
43elpw2 4956 . 2 (𝑆 ∈ 𝒫 𝑉𝑆𝑉)
5 ocvfval.i . . . . . 6 , = (·𝑖𝑊)
6 ocvfval.f . . . . . 6 𝐹 = (Scalar‘𝑊)
7 ocvfval.z . . . . . 6 0 = (0g𝐹)
8 ocvfval.o . . . . . 6 = (ocv‘𝑊)
91, 5, 6, 7, 8ocvfval 20226 . . . . 5 (𝑊 ∈ V → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
109fveq1d 6334 . . . 4 (𝑊 ∈ V → ( 𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
11 raleq 3286 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 ))
1211rabbidv 3338 . . . . 5 (𝑠 = 𝑆 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
13 eqid 2770 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
143rabex 4943 . . . . 5 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ∈ V
1512, 13, 14fvmpt 6424 . . . 4 (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
1610, 15sylan9eq 2824 . . 3 ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
17 0fv 6368 . . . . 5 (∅‘𝑆) = ∅
18 fvprc 6326 . . . . . . 7 𝑊 ∈ V → (ocv‘𝑊) = ∅)
198, 18syl5eq 2816 . . . . . 6 𝑊 ∈ V → = ∅)
2019fveq1d 6334 . . . . 5 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
21 ssrab2 3834 . . . . . 6 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
22 fvprc 6326 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
231, 22syl5eq 2816 . . . . . 6 𝑊 ∈ V → 𝑉 = ∅)
24 sseq0 4117 . . . . . 6 (({𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉𝑉 = ∅) → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2521, 23, 24sylancr 567 . . . . 5 𝑊 ∈ V → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2617, 20, 253eqtr4a 2830 . . . 4 𝑊 ∈ V → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2726adantr 466 . . 3 ((¬ 𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2816, 27pm2.61ian 795 . 2 (𝑆 ∈ 𝒫 𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
294, 28sylbir 225 1 (𝑆𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1630  wcel 2144  wral 3060  {crab 3064  Vcvv 3349  wss 3721  c0 4061  𝒫 cpw 4295  cmpt 4861  cfv 6031  (class class class)co 6792  Basecbs 16063  Scalarcsca 16151  ·𝑖cip 16153  0gc0g 16307  ocvcocv 20220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-ocv 20223
This theorem is referenced by:  elocv  20228  ocv0  20237  csscld  23266  hlhilocv  37760
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