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Theorem ocvval 20982
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 6848 . . 3 𝑉 ∈ V
32elpw2 5297 . 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 20981 . . . . 5 (𝑊 ∈ V → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
98fveq1d 6836 . . . 4 (𝑊 ∈ V → ( 𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10 raleq 3307 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 ))
1110rabbidv 3413 . . . . 5 (𝑠 = 𝑆 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
12 eqid 2737 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
132rabex 5284 . . . . 5 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ∈ V
1411, 12, 13fvmpt 6940 . . . 4 (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
159, 14sylan9eq 2797 . . 3 ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
16 0fv 6878 . . . . 5 (∅‘𝑆) = ∅
17 fvprc 6826 . . . . . . 7 𝑊 ∈ V → (ocv‘𝑊) = ∅)
187, 17eqtrid 2789 . . . . . 6 𝑊 ∈ V → = ∅)
1918fveq1d 6836 . . . . 5 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
20 ssrab2 4032 . . . . . 6 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21 fvprc 6826 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
221, 21eqtrid 2789 . . . . . 6 𝑊 ∈ V → 𝑉 = ∅)
23 sseq0 4354 . . . . . 6 (({𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉𝑉 = ∅) → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2420, 22, 23sylancr 588 . . . . 5 𝑊 ∈ V → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2516, 19, 243eqtr4a 2803 . . . 4 𝑊 ∈ V → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2625adantr 482 . . 3 ((¬ 𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2715, 26pm2.61ian 810 . 2 (𝑆 ∈ 𝒫 𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
283, 27sylbir 234 1 (𝑆𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wcel 2106  wral 3062  {crab 3405  Vcvv 3443  wss 3905  c0 4277  𝒫 cpw 4555  cmpt 5183  cfv 6488  (class class class)co 7346  Basecbs 17014  Scalarcsca 17067  ·𝑖cip 17069  0gc0g 17252  ocvcocv 20975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5251  ax-nul 5258  ax-pow 5315  ax-pr 5379  ax-un 7659
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4278  df-if 4482  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4861  df-br 5101  df-opab 5163  df-mpt 5184  df-id 5525  df-xp 5633  df-rel 5634  df-cnv 5635  df-co 5636  df-dm 5637  df-rn 5638  df-res 5639  df-ima 5640  df-iota 6440  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ov 7349  df-ocv 20978
This theorem is referenced by:  elocv  20983  ocv0  20992  csscld  24523  hlhilocv  40280
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