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Theorem ocvval 21087
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 6861 . . 3 𝑉 ∈ V
32elpw2 5307 . 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 21086 . . . . 5 (π‘Š ∈ V β†’ βŠ₯ = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
98fveq1d 6849 . . . 4 (π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = ((𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })β€˜π‘†))
10 raleq 3312 . . . . . 6 (𝑠 = 𝑆 β†’ (βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 ↔ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 ))
1110rabbidv 3418 . . . . 5 (𝑠 = 𝑆 β†’ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 } = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
12 eqid 2737 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })
132rabex 5294 . . . . 5 {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } ∈ V
1411, 12, 13fvmpt 6953 . . . 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 2797 . . 3 ((π‘Š ∈ V ∧ 𝑆 ∈ 𝒫 𝑉) β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
16 0fv 6891 . . . . 5 (βˆ…β€˜π‘†) = βˆ…
17 fvprc 6839 . . . . . . 7 (Β¬ π‘Š ∈ V β†’ (ocvβ€˜π‘Š) = βˆ…)
187, 17eqtrid 2789 . . . . . 6 (Β¬ π‘Š ∈ V β†’ βŠ₯ = βˆ…)
1918fveq1d 6849 . . . . 5 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = (βˆ…β€˜π‘†))
20 ssrab2 4042 . . . . . 6 {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } βŠ† 𝑉
21 fvprc 6839 . . . . . . 7 (Β¬ π‘Š ∈ V β†’ (Baseβ€˜π‘Š) = βˆ…)
221, 21eqtrid 2789 . . . . . 6 (Β¬ π‘Š ∈ V β†’ 𝑉 = βˆ…)
23 sseq0 4364 . . . . . 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 588 . . . . 5 (Β¬ π‘Š ∈ V β†’ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 } = βˆ…)
2516, 19, 243eqtr4a 2803 . . . 4 (Β¬ π‘Š ∈ V β†’ ( βŠ₯ β€˜π‘†) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑆 (π‘₯ , 𝑦) = 0 })
2625adantr 482 . . 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 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  Scalarcsca 17143  Β·π‘–cip 17145  0gc0g 17328  ocvcocv 21080
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-ocv 21083
This theorem is referenced by:  elocv  21088  ocv0  21097  csscld  24629  hlhilocv  40453
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