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Theorem elocv 20361
 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 6677 . . . . 5 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ dom )
2 n0i 4249 . . . . . . . . 9 (𝐴 ∈ ( 𝑆) → ¬ ( 𝑆) = ∅)
3 ocvfval.o . . . . . . . . . . . 12 = (ocv‘𝑊)
4 fvprc 6638 . . . . . . . . . . . 12 𝑊 ∈ V → (ocv‘𝑊) = ∅)
53, 4syl5eq 2845 . . . . . . . . . . 11 𝑊 ∈ V → = ∅)
65fveq1d 6647 . . . . . . . . . 10 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
7 0fv 6684 . . . . . . . . . 10 (∅‘𝑆) = ∅
86, 7eqtrdi 2849 . . . . . . . . 9 𝑊 ∈ V → ( 𝑆) = ∅)
92, 8nsyl2 143 . . . . . . . 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 20359 . . . . . . . 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 5738 . . . . . 6 (𝐴 ∈ ( 𝑆) → dom = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
1710fvexi 6659 . . . . . . . 8 𝑉 ∈ V
1817rabex 5199 . . . . . . 7 {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19 eqid 2798 . . . . . . 7 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 })
2018, 19dmmpti 6464 . . . . . 6 dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
2116, 20eqtrdi 2849 . . . . 5 (𝐴 ∈ ( 𝑆) → dom = 𝒫 𝑉)
221, 21eleqtrd 2892 . . . 4 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ 𝒫 𝑉)
2322elpwid 4508 . . 3 (𝐴 ∈ ( 𝑆) → 𝑆𝑉)
2410, 11, 12, 13, 3ocvval 20360 . . . . 5 (𝑆𝑉 → ( 𝑆) = {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 })
2524eleq2d 2875 . . . 4 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ 𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 }))
26 oveq1 7142 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
2726eqeq1d 2800 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
2827ralbidv 3162 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
2928elrab 3628 . . . 4 (𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
3025, 29syl6bb 290 . . 3 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3123, 30biadanii 821 . 2 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
32 3anass 1092 . 2 ((𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3331, 32bitr4i 281 1 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   ↦ cmpt 5110  dom cdm 5519  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  Scalarcsca 16562  ·𝑖cip 16564  0gc0g 16707  ocvcocv 20353 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-ocv 20356 This theorem is referenced by:  ocvi  20362  ocvss  20363  ocvocv  20364  ocvlss  20365  ocv2ss  20366  unocv  20373  iunocv  20374  obselocv  20421  clsocv  23861  pjthlem2  24049
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