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Theorem ocvval 21599
Description: Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised 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
ocvval (𝑆 βŠ† 𝑉 β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
Distinct variable groups:   π‘₯,𝑦, 0   π‘₯,𝑉,𝑦   π‘₯,π‘Š,𝑦   π‘₯, , ,𝑦   π‘₯,𝑆,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦)

Proof of Theorem ocvval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ocvfval.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
21fvexi 6911 . . 3 𝑉 ∈ V
32elpw2 5347 . 2 (𝑆 ∈ 𝒫 𝑉 ↔ 𝑆 βŠ† 𝑉)
4 ocvfval.i . . . . . 6 , = (Β·π‘–β€˜π‘Š)
5 ocvfval.f . . . . . 6 𝐹 = (Scalarβ€˜π‘Š)
6 ocvfval.z . . . . . 6 0 = (0gβ€˜πΉ)
7 ocvfval.o . . . . . 6 βŠ₯ = (ocvβ€˜π‘Š)
81, 4, 5, 6, 7ocvfval 21598 . . . . 5 (π‘Š ∈ V β†’ βŠ₯ = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
98fveq1d 6899 . . . 4 (π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = ((𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })β€˜π‘†))
10 raleq 3319 . . . . . 6 (𝑠 = 𝑆 β†’ (βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 ↔ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 ))
1110rabbidv 3437 . . . . 5 (𝑠 = 𝑆 β†’ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 } = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
12 eqid 2728 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })
132rabex 5334 . . . . 5 {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } ∈ V
1411, 12, 13fvmpt 7005 . . . 4 (𝑆 ∈ 𝒫 𝑉 β†’ ((𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
159, 14sylan9eq 2788 . . 3 ((π‘Š ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
16 0fv 6941 . . . . 5 (βˆ…β€˜π‘†) = βˆ…
17 fvprc 6889 . . . . . . 7 (Β¬ π‘Š ∈ V β†’ (ocvβ€˜π‘Š) = βˆ…)
187, 17eqtrid 2780 . . . . . 6 (Β¬ π‘Š ∈ V β†’ βŠ₯ = βˆ…)
1918fveq1d 6899 . . . . 5 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = (βˆ…β€˜π‘†))
20 ssrab2 4075 . . . . . 6 {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } βŠ† 𝑉
21 fvprc 6889 . . . . . . 7 (Β¬ π‘Š ∈ V β†’ (Baseβ€˜π‘Š) = βˆ…)
221, 21eqtrid 2780 . . . . . 6 (Β¬ π‘Š ∈ V β†’ 𝑉 = βˆ…)
23 sseq0 4400 . . . . . 6 (({π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } βŠ† 𝑉 ∧ 𝑉 = βˆ…) β†’ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } = βˆ…)
2420, 22, 23sylancr 586 . . . . 5 (Β¬ π‘Š ∈ V β†’ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } = βˆ…)
2516, 19, 243eqtr4a 2794 . . . 4 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
2625adantr 480 . . 3 ((Β¬ π‘Š ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
2715, 26pm2.61ian 811 . 2 (𝑆 ∈ 𝒫 𝑉 β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
283, 27sylbir 234 1 (𝑆 βŠ† 𝑉 β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  {crab 3429  Vcvv 3471   βŠ† wss 3947  βˆ…c0 4323  π’« cpw 4603   ↦ cmpt 5231  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  Scalarcsca 17236  Β·π‘–cip 17238  0gc0g 17421  ocvcocv 21592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-ocv 21595
This theorem is referenced by:  elocv  21600  ocv0  21609  csscld  25190  hlhilocv  41434
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