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Theorem elocv 21221
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 = (Baseβ€˜π‘Š)
ocvfval.i , = (Β·π‘–β€˜π‘Š)
ocvfval.f 𝐹 = (Scalarβ€˜π‘Š)
ocvfval.z 0 = (0gβ€˜πΉ)
ocvfval.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
elocv (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
Distinct variable groups:   π‘₯, 0   π‘₯,𝐴   π‘₯,𝑉   π‘₯,π‘Š   π‘₯, ,   π‘₯,𝑆
Allowed substitution hints:   𝐹(π‘₯)   βŠ₯ (π‘₯)

Proof of Theorem elocv
Dummy variables 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6929 . . . . 5 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ 𝑆 ∈ dom βŠ₯ )
2 n0i 4334 . . . . . . . . 9 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ Β¬ ( βŠ₯ β€˜π‘†) = βˆ…)
3 ocvfval.o . . . . . . . . . . . 12 βŠ₯ = (ocvβ€˜π‘Š)
4 fvprc 6884 . . . . . . . . . . . 12 (Β¬ π‘Š ∈ V β†’ (ocvβ€˜π‘Š) = βˆ…)
53, 4eqtrid 2785 . . . . . . . . . . 11 (Β¬ π‘Š ∈ V β†’ βŠ₯ = βˆ…)
65fveq1d 6894 . . . . . . . . . 10 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = (βˆ…β€˜π‘†))
7 0fv 6936 . . . . . . . . . 10 (βˆ…β€˜π‘†) = βˆ…
86, 7eqtrdi 2789 . . . . . . . . 9 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = βˆ…)
92, 8nsyl2 141 . . . . . . . 8 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ π‘Š ∈ V)
10 ocvfval.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
11 ocvfval.i . . . . . . . . 9 , = (Β·π‘–β€˜π‘Š)
12 ocvfval.f . . . . . . . . 9 𝐹 = (Scalarβ€˜π‘Š)
13 ocvfval.z . . . . . . . . 9 0 = (0gβ€˜πΉ)
1410, 11, 12, 13, 3ocvfval 21219 . . . . . . . 8 (π‘Š ∈ V β†’ βŠ₯ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 }))
159, 14syl 17 . . . . . . 7 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ βŠ₯ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 }))
1615dmeqd 5906 . . . . . 6 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ dom βŠ₯ = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 }))
1710fvexi 6906 . . . . . . . 8 𝑉 ∈ V
1817rabex 5333 . . . . . . 7 {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 } ∈ V
19 eqid 2733 . . . . . . 7 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 })
2018, 19dmmpti 6695 . . . . . 6 dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 }) = 𝒫 𝑉
2116, 20eqtrdi 2789 . . . . 5 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ dom βŠ₯ = 𝒫 𝑉)
221, 21eleqtrd 2836 . . . 4 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ 𝑆 ∈ 𝒫 𝑉)
2322elpwid 4612 . . 3 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ 𝑆 βŠ† 𝑉)
2410, 11, 12, 13, 3ocvval 21220 . . . . 5 (𝑆 βŠ† 𝑉 β†’ ( βŠ₯ β€˜π‘†) = {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑆 (𝑦 , π‘₯) = 0 })
2524eleq2d 2820 . . . 4 (𝑆 βŠ† 𝑉 β†’ (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ 𝐴 ∈ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑆 (𝑦 , π‘₯) = 0 }))
26 oveq1 7416 . . . . . . 7 (𝑦 = 𝐴 β†’ (𝑦 , π‘₯) = (𝐴 , π‘₯))
2726eqeq1d 2735 . . . . . 6 (𝑦 = 𝐴 β†’ ((𝑦 , π‘₯) = 0 ↔ (𝐴 , π‘₯) = 0 ))
2827ralbidv 3178 . . . . 5 (𝑦 = 𝐴 β†’ (βˆ€π‘₯ ∈ 𝑆 (𝑦 , π‘₯) = 0 ↔ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
2928elrab 3684 . . . 4 (𝐴 ∈ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑆 (𝑦 , π‘₯) = 0 } ↔ (𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
3025, 29bitrdi 287 . . 3 (𝑆 βŠ† 𝑉 β†’ (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ (𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 )))
3123, 30biadanii 821 . 2 (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 )))
32 3anass 1096 . 2 ((𝑆 βŠ† 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ) ↔ (𝑆 βŠ† 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 )))
3331, 32bitr4i 278 1 (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603   ↦ cmpt 5232  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200  Β·π‘–cip 17202  0gc0g 17385  ocvcocv 21213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-ocv 21216
This theorem is referenced by:  ocvi  21222  ocvss  21223  ocvocv  21224  ocvlss  21225  ocv2ss  21226  unocv  21233  iunocv  21234  obselocv  21283  clsocv  24767  pjthlem2  24955
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