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Theorem ocvval 21071
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 6856 . . 3 𝑉 ∈ V
32elpw2 5302 . 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 21070 . . . . 5 (𝑊 ∈ V → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
98fveq1d 6844 . . . 4 (𝑊 ∈ V → ( 𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
10 raleq 3309 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 ))
1110rabbidv 3415 . . . . 5 (𝑠 = 𝑆 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
12 eqid 2736 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
132rabex 5289 . . . . 5 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ∈ V
1411, 12, 13fvmpt 6948 . . . 4 (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
159, 14sylan9eq 2796 . . 3 ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
16 0fv 6886 . . . . 5 (∅‘𝑆) = ∅
17 fvprc 6834 . . . . . . 7 𝑊 ∈ V → (ocv‘𝑊) = ∅)
187, 17eqtrid 2788 . . . . . 6 𝑊 ∈ V → = ∅)
1918fveq1d 6844 . . . . 5 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
20 ssrab2 4037 . . . . . 6 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
21 fvprc 6834 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
221, 21eqtrid 2788 . . . . . 6 𝑊 ∈ V → 𝑉 = ∅)
23 sseq0 4359 . . . . . 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 2802 . . . 4 𝑊 ∈ V → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2625adantr 481 . . 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 3064  {crab 3407  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560  cmpt 5188  cfv 6496  (class class class)co 7357  Basecbs 17083  Scalarcsca 17136  ·𝑖cip 17138  0gc0g 17321  ocvcocv 21064
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-ocv 21067
This theorem is referenced by:  elocv  21072  ocv0  21081  csscld  24613  hlhilocv  40424
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