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Theorem elocv 21212
Description: Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed 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
elocv (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝑉   𝑥,𝑊   𝑥, ,   𝑥,𝑆
Allowed substitution hints:   𝐹(𝑥)   (𝑥)

Proof of Theorem elocv
Dummy variables 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6925 . . . . 5 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ dom )
2 n0i 4332 . . . . . . . . 9 (𝐴 ∈ ( 𝑆) → ¬ ( 𝑆) = ∅)
3 ocvfval.o . . . . . . . . . . . 12 = (ocv‘𝑊)
4 fvprc 6880 . . . . . . . . . . . 12 𝑊 ∈ V → (ocv‘𝑊) = ∅)
53, 4eqtrid 2784 . . . . . . . . . . 11 𝑊 ∈ V → = ∅)
65fveq1d 6890 . . . . . . . . . 10 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
7 0fv 6932 . . . . . . . . . 10 (∅‘𝑆) = ∅
86, 7eqtrdi 2788 . . . . . . . . 9 𝑊 ∈ V → ( 𝑆) = ∅)
92, 8nsyl2 141 . . . . . . . 8 (𝐴 ∈ ( 𝑆) → 𝑊 ∈ V)
10 ocvfval.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
11 ocvfval.i . . . . . . . . 9 , = (·𝑖𝑊)
12 ocvfval.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
13 ocvfval.z . . . . . . . . 9 0 = (0g𝐹)
1410, 11, 12, 13, 3ocvfval 21210 . . . . . . . 8 (𝑊 ∈ V → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
159, 14syl 17 . . . . . . 7 (𝐴 ∈ ( 𝑆) → = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
1615dmeqd 5903 . . . . . 6 (𝐴 ∈ ( 𝑆) → dom = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
1710fvexi 6902 . . . . . . . 8 𝑉 ∈ V
1817rabex 5331 . . . . . . 7 {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19 eqid 2732 . . . . . . 7 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 })
2018, 19dmmpti 6691 . . . . . 6 dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
2116, 20eqtrdi 2788 . . . . 5 (𝐴 ∈ ( 𝑆) → dom = 𝒫 𝑉)
221, 21eleqtrd 2835 . . . 4 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ 𝒫 𝑉)
2322elpwid 4610 . . 3 (𝐴 ∈ ( 𝑆) → 𝑆𝑉)
2410, 11, 12, 13, 3ocvval 21211 . . . . 5 (𝑆𝑉 → ( 𝑆) = {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 })
2524eleq2d 2819 . . . 4 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ 𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 }))
26 oveq1 7412 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
2726eqeq1d 2734 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
2827ralbidv 3177 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
2928elrab 3682 . . . 4 (𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
3025, 29bitrdi 286 . . 3 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3123, 30biadanii 820 . 2 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
32 3anass 1095 . 2 ((𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3331, 32bitr4i 277 1 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  {crab 3432  Vcvv 3474  wss 3947  c0 4321  𝒫 cpw 4601  cmpt 5230  dom cdm 5675  cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196  ·𝑖cip 17198  0gc0g 17381  ocvcocv 21204
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-ocv 21207
This theorem is referenced by:  ocvi  21213  ocvss  21214  ocvocv  21215  ocvlss  21216  ocv2ss  21217  unocv  21224  iunocv  21225  obselocv  21274  clsocv  24758  pjthlem2  24946
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