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Theorem elocv 20229
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 6361 . . . . 5 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ dom )
2 n0i 4068 . . . . . . . . 9 (𝐴 ∈ ( 𝑆) → ¬ ( 𝑆) = ∅)
3 ocvfval.o . . . . . . . . . . . 12 = (ocv‘𝑊)
4 fvprc 6326 . . . . . . . . . . . 12 𝑊 ∈ V → (ocv‘𝑊) = ∅)
53, 4syl5eq 2817 . . . . . . . . . . 11 𝑊 ∈ V → = ∅)
65fveq1d 6334 . . . . . . . . . 10 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
7 0fv 6368 . . . . . . . . . 10 (∅‘𝑆) = ∅
86, 7syl6eq 2821 . . . . . . . . 9 𝑊 ∈ V → ( 𝑆) = ∅)
92, 8nsyl2 144 . . . . . . . 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 20227 . . . . . . . 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 5464 . . . . . 6 (𝐴 ∈ ( 𝑆) → dom = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
17 fvex 6342 . . . . . . . . 9 (Base‘𝑊) ∈ V
1810, 17eqeltri 2846 . . . . . . . 8 𝑉 ∈ V
1918rabex 4946 . . . . . . 7 {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 } ∈ V
20 eqid 2771 . . . . . . 7 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 })
2119, 20dmmpti 6163 . . . . . 6 dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
2216, 21syl6eq 2821 . . . . 5 (𝐴 ∈ ( 𝑆) → dom = 𝒫 𝑉)
231, 22eleqtrd 2852 . . . 4 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ 𝒫 𝑉)
2423elpwid 4309 . . 3 (𝐴 ∈ ( 𝑆) → 𝑆𝑉)
2510, 11, 12, 13, 3ocvval 20228 . . . . 5 (𝑆𝑉 → ( 𝑆) = {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 })
2625eleq2d 2836 . . . 4 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ 𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 }))
27 oveq1 6800 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
2827eqeq1d 2773 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
2928ralbidv 3135 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
3029elrab 3515 . . . 4 (𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
3126, 30syl6bb 276 . . 3 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3224, 31biadan2 820 . 2 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
33 3anass 1080 . 2 ((𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3432, 33bitr4i 267 1 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297  cmpt 4863  dom cdm 5249  cfv 6031  (class class class)co 6793  Basecbs 16064  Scalarcsca 16152  ·𝑖cip 16154  0gc0g 16308  ocvcocv 20221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  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 6796  df-ocv 20224
This theorem is referenced by:  ocvi  20230  ocvss  20231  ocvocv  20232  ocvlss  20233  ocv2ss  20234  unocv  20241  iunocv  20242  obselocv  20289  clsocv  23268  pjthlem2  23428
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