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Theorem elocv 21095
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 6883 . . . . 5 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ 𝑆 ∈ dom βŠ₯ )
2 n0i 4297 . . . . . . . . 9 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ Β¬ ( βŠ₯ β€˜π‘†) = βˆ…)
3 ocvfval.o . . . . . . . . . . . 12 βŠ₯ = (ocvβ€˜π‘Š)
4 fvprc 6838 . . . . . . . . . . . 12 (Β¬ π‘Š ∈ V β†’ (ocvβ€˜π‘Š) = βˆ…)
53, 4eqtrid 2785 . . . . . . . . . . 11 (Β¬ π‘Š ∈ V β†’ βŠ₯ = βˆ…)
65fveq1d 6848 . . . . . . . . . 10 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = (βˆ…β€˜π‘†))
7 0fv 6890 . . . . . . . . . 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 21093 . . . . . . . 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 5865 . . . . . 6 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ dom βŠ₯ = dom (𝑠 ∈ 𝒫 𝑉 ↦ {𝑦 ∈ 𝑉 ∣ βˆ€π‘₯ ∈ 𝑠 (𝑦 , π‘₯) = 0 }))
1710fvexi 6860 . . . . . . . 8 𝑉 ∈ V
1817rabex 5293 . . . . . . 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 6649 . . . . . 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 4573 . . 3 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ 𝑆 βŠ† 𝑉)
2410, 11, 12, 13, 3ocvval 21094 . . . . 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 7368 . . . . . . 7 (𝑦 = 𝐴 β†’ (𝑦 , π‘₯) = (𝐴 , π‘₯))
2726eqeq1d 2735 . . . . . 6 (𝑦 = 𝐴 β†’ ((𝑦 , π‘₯) = 0 ↔ (𝐴 , π‘₯) = 0 ))
2827ralbidv 3171 . . . . 5 (𝑦 = 𝐴 β†’ (βˆ€π‘₯ ∈ 𝑆 (𝑦 , π‘₯) = 0 ↔ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
2928elrab 3649 . . . 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 3061  {crab 3406  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564   ↦ cmpt 5192  dom cdm 5637  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  Scalarcsca 17144  Β·π‘–cip 17146  0gc0g 17329  ocvcocv 21087
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-ocv 21090
This theorem is referenced by:  ocvi  21096  ocvss  21097  ocvocv  21098  ocvlss  21099  ocv2ss  21100  unocv  21107  iunocv  21108  obselocv  21157  clsocv  24637  pjthlem2  24825
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