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Theorem ocvval 19939
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 6163 . . . 4 (Base‘𝑊) ∈ V
31, 2eqeltri 2694 . . 3 𝑉 ∈ V
43elpw2 4793 . 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 19938 . . . . 5 (𝑊 ∈ V → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
109fveq1d 6155 . . . 4 (𝑊 ∈ V → ( 𝑆) = ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆))
11 raleq 3130 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 , 𝑦) = 0 ↔ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 ))
1211rabbidv 3180 . . . . 5 (𝑠 = 𝑆 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
13 eqid 2621 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
143rabex 4778 . . . . 5 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ∈ V
1512, 13, 14fvmpt 6244 . . . 4 (𝑆 ∈ 𝒫 𝑉 → ((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })‘𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
1610, 15sylan9eq 2675 . . 3 ((𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
17 0fv 6189 . . . . 5 (∅‘𝑆) = ∅
18 fvprc 6147 . . . . . . 7 𝑊 ∈ V → (ocv‘𝑊) = ∅)
198, 18syl5eq 2667 . . . . . 6 𝑊 ∈ V → = ∅)
2019fveq1d 6155 . . . . 5 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
21 ssrab2 3671 . . . . . 6 {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
22 fvprc 6147 . . . . . . 7 𝑊 ∈ V → (Base‘𝑊) = ∅)
231, 22syl5eq 2667 . . . . . 6 𝑊 ∈ V → 𝑉 = ∅)
24 sseq0 3952 . . . . . 6 (({𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } ⊆ 𝑉𝑉 = ∅) → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2521, 23, 24sylancr 694 . . . . 5 𝑊 ∈ V → {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 } = ∅)
2617, 20, 253eqtr4a 2681 . . . 4 𝑊 ∈ V → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2726adantr 481 . . 3 ((¬ 𝑊 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })
2816, 27pm2.61ian 830 . 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 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  wss 3559  c0 3896  𝒫 cpw 4135  cmpt 4678  cfv 5852  (class class class)co 6610  Basecbs 15788  Scalarcsca 15872  ·𝑖cip 15874  0gc0g 16028  ocvcocv 19932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-ocv 19935
This theorem is referenced by:  elocv  19940  ocv0  19949  csscld  22967  hlhilocv  36756
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