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Theorem ocvfval 20942
Description: The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised 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
ocvfval (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Distinct variable groups:   𝑥,𝑠,𝑦, 0   𝑉,𝑠,𝑥,𝑦   𝑊,𝑠,𝑥,𝑦   , ,𝑠,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑠)   (𝑥,𝑦,𝑠)   𝑋(𝑥,𝑦,𝑠)

Proof of Theorem ocvfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ocvfval.o . 2 = (ocv‘𝑊)
2 elex 3459 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6809 . . . . . . 7 ( = 𝑊 → (Base‘) = (Base‘𝑊))
4 ocvfval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4eqtr4di 2795 . . . . . 6 ( = 𝑊 → (Base‘) = 𝑉)
65pweqd 4560 . . . . 5 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
7 fveq2 6809 . . . . . . . . . 10 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
8 ocvfval.i . . . . . . . . . 10 , = (·𝑖𝑊)
97, 8eqtr4di 2795 . . . . . . . . 9 ( = 𝑊 → (·𝑖) = , )
109oveqd 7330 . . . . . . . 8 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
11 fveq2 6809 . . . . . . . . . . 11 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
12 ocvfval.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1311, 12eqtr4di 2795 . . . . . . . . . 10 ( = 𝑊 → (Scalar‘) = 𝐹)
1413fveq2d 6813 . . . . . . . . 9 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
15 ocvfval.z . . . . . . . . 9 0 = (0g𝐹)
1614, 15eqtr4di 2795 . . . . . . . 8 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
1710, 16eqeq12d 2753 . . . . . . 7 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ (𝑥 , 𝑦) = 0 ))
1817ralbidv 3171 . . . . . 6 ( = 𝑊 → (∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 ))
195, 18rabeqbidv 3419 . . . . 5 ( = 𝑊 → {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))} = {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
206, 19mpteq12dv 5176 . . . 4 ( = 𝑊 → (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
21 df-ocv 20939 . . . 4 ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))
22 eqid 2737 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
234fvexi 6823 . . . . . . . 8 𝑉 ∈ V
24 ssrab2 4023 . . . . . . . 8 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
2523, 24elpwi2 5283 . . . . . . 7 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉
2625a1i 11 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉)
2722, 26fmpti 7023 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
2823pwex 5316 . . . . 5 𝒫 𝑉 ∈ V
29 fex2 7823 . . . . 5 (((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉 ∧ 𝒫 𝑉 ∈ V ∧ 𝒫 𝑉 ∈ V) → (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V)
3027, 28, 28, 29mp3an 1460 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
3120, 21, 30fvmpt 6912 . . 3 (𝑊 ∈ V → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
322, 31syl 17 . 2 (𝑊𝑋 → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
331, 32eqtrid 2789 1 (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  wral 3062  {crab 3404  Vcvv 3441  𝒫 cpw 4543  cmpt 5168  wf 6459  cfv 6463  (class class class)co 7313  Basecbs 16979  Scalarcsca 17032  ·𝑖cip 17034  0gc0g 17217  ocvcocv 20936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fv 6471  df-ov 7316  df-ocv 20939
This theorem is referenced by:  ocvval  20943  elocv  20944
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