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Theorem elocv 20740
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 6695 . . . . 5 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ dom )
2 n0i 4296 . . . . . . . . 9 (𝐴 ∈ ( 𝑆) → ¬ ( 𝑆) = ∅)
3 ocvfval.o . . . . . . . . . . . 12 = (ocv‘𝑊)
4 fvprc 6656 . . . . . . . . . . . 12 𝑊 ∈ V → (ocv‘𝑊) = ∅)
53, 4syl5eq 2865 . . . . . . . . . . 11 𝑊 ∈ V → = ∅)
65fveq1d 6665 . . . . . . . . . 10 𝑊 ∈ V → ( 𝑆) = (∅‘𝑆))
7 0fv 6702 . . . . . . . . . 10 (∅‘𝑆) = ∅
86, 7syl6eq 2869 . . . . . . . . 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 20738 . . . . . . . 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 5767 . . . . . 6 (𝐴 ∈ ( 𝑆) → dom = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }))
1710fvexi 6677 . . . . . . . 8 𝑉 ∈ V
1817rabex 5226 . . . . . . 7 {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 } ∈ V
19 eqid 2818 . . . . . . 7 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 })
2018, 19dmmpti 6485 . . . . . 6 dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦𝑉 ∣ ∀𝑥𝑠 (𝑦 , 𝑥) = 0 }) = 𝒫 𝑉
2116, 20syl6eq 2869 . . . . 5 (𝐴 ∈ ( 𝑆) → dom = 𝒫 𝑉)
221, 21eleqtrd 2912 . . . 4 (𝐴 ∈ ( 𝑆) → 𝑆 ∈ 𝒫 𝑉)
2322elpwid 4549 . . 3 (𝐴 ∈ ( 𝑆) → 𝑆𝑉)
2410, 11, 12, 13, 3ocvval 20739 . . . . 5 (𝑆𝑉 → ( 𝑆) = {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 })
2524eleq2d 2895 . . . 4 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ 𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 }))
26 oveq1 7152 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 , 𝑥) = (𝐴 , 𝑥))
2726eqeq1d 2820 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 , 𝑥) = 0 ↔ (𝐴 , 𝑥) = 0 ))
2827ralbidv 3194 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑆 (𝑦 , 𝑥) = 0 ↔ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
2928elrab 3677 . . . 4 (𝐴 ∈ {𝑦𝑉 ∣ ∀𝑥𝑆 (𝑦 , 𝑥) = 0 } ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
3025, 29syl6bb 288 . . 3 (𝑆𝑉 → (𝐴 ∈ ( 𝑆) ↔ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3123, 30biadanii 818 . 2 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
32 3anass 1087 . 2 ((𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ) ↔ (𝑆𝑉 ∧ (𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )))
3331, 32bitr4i 279 1 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535  cmpt 5137  dom cdm 5548  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556  ·𝑖cip 16558  0gc0g 16701  ocvcocv 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-ocv 20735
This theorem is referenced by:  ocvi  20741  ocvss  20742  ocvocv  20743  ocvlss  20744  ocv2ss  20745  unocv  20752  iunocv  20753  obselocv  20800  clsocv  23780  pjthlem2  23968
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