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Theorem elocv 20212
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 6379 . . . . 5 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ dom )
2 n0i 4061 . . . . . . . . 9 (𝐴 ∈ ( 𝑆) → ¬ ( 𝑆) = ∅)
3 ocvfval.o . . . . . . . . . . . 12 = (ocv‘𝑊)
4 fvprc 6344 . . . . . . . . . . . 12 𝑊 ∈ V → (ocv‘𝑊) = ∅)
53, 4syl5eq 2804 . . . . . . . . . . 11 𝑊 ∈ V → = ∅)
65fveq1d 6352 . . . . . . . . . 10 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
7 0fv 6386 . . . . . . . . . 10 (∅‘𝑆) = ∅
86, 7syl6eq 2808 . . . . . . . . 9 𝑊 ∈ V → ( 𝑆) = ∅)
92, 8nsyl2 142 . . . . . . . 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 20210 . . . . . . . 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 5479 . . . . . 6 (𝐴 ∈ ( 𝑆) → dom = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
17 fvex 6360 . . . . . . . . 9 (Base‘𝑊) ∈ V
1810, 17eqeltri 2833 . . . . . . . 8 𝑉 ∈ V
1918rabex 4962 . . . . . . 7 {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 } ∈ V
20 eqid 2758 . . . . . . 7 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 })
2119, 20dmmpti 6182 . . . . . 6 dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
2216, 21syl6eq 2808 . . . . 5 (𝐴 ∈ ( 𝑆) → dom = 𝒫 𝑉)
231, 22eleqtrd 2839 . . . 4 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ 𝒫 𝑉)
2423elpwid 4312 . . 3 (𝐴 ∈ ( 𝑆) → 𝑆𝑉)
2510, 11, 12, 13, 3ocvval 20211 . . . . 5 (𝑆𝑉 → ( 𝑆) = {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 })
2625eleq2d 2823 . . . 4 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ 𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 }))
27 oveq1 6818 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
2827eqeq1d 2760 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
2928ralbidv 3122 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
3029elrab 3502 . . . 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 677 . 2 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
33 3anass 1081 . 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 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  {crab 3052  Vcvv 3338  wss 3713  c0 4056  𝒫 cpw 4300  cmpt 4879  dom cdm 5264  cfv 6047  (class class class)co 6811  Basecbs 16057  Scalarcsca 16144  ·𝑖cip 16146  0gc0g 16300  ocvcocv 20204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fv 6055  df-ov 6814  df-ocv 20207
This theorem is referenced by:  ocvi  20213  ocvss  20214  ocvocv  20215  ocvlss  20216  ocv2ss  20217  unocv  20224  iunocv  20225  obselocv  20272  clsocv  23247  pjthlem2  23407
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