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Theorem ocvfval 19929
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 3198 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6148 . . . . . . 7 ( = 𝑊 → (Base‘) = (Base‘𝑊))
4 ocvfval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2673 . . . . . 6 ( = 𝑊 → (Base‘) = 𝑉)
65pweqd 4135 . . . . 5 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
7 fveq2 6148 . . . . . . . . . 10 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
8 ocvfval.i . . . . . . . . . 10 , = (·𝑖𝑊)
97, 8syl6eqr 2673 . . . . . . . . 9 ( = 𝑊 → (·𝑖) = , )
109oveqd 6621 . . . . . . . 8 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
11 fveq2 6148 . . . . . . . . . . 11 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
12 ocvfval.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1311, 12syl6eqr 2673 . . . . . . . . . 10 ( = 𝑊 → (Scalar‘) = 𝐹)
1413fveq2d 6152 . . . . . . . . 9 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
15 ocvfval.z . . . . . . . . 9 0 = (0g𝐹)
1614, 15syl6eqr 2673 . . . . . . . 8 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
1710, 16eqeq12d 2636 . . . . . . 7 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ (𝑥 , 𝑦) = 0 ))
1817ralbidv 2980 . . . . . 6 ( = 𝑊 → (∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 ))
195, 18rabeqbidv 3181 . . . . 5 ( = 𝑊 → {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))} = {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
206, 19mpteq12dv 4693 . . . 4 ( = 𝑊 → (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
21 df-ocv 19926 . . . 4 ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))
22 eqid 2621 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
23 ssrab2 3666 . . . . . . . 8 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
24 fvex 6158 . . . . . . . . . 10 (Base‘𝑊) ∈ V
254, 24eqeltri 2694 . . . . . . . . 9 𝑉 ∈ V
2625elpw2 4788 . . . . . . . 8 ({𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉 ↔ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉)
2723, 26mpbir 221 . . . . . . 7 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉
2827a1i 11 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 → {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉)
2922, 28fmpti 6339 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
3025pwex 4808 . . . . 5 𝒫 𝑉 ∈ V
31 fex2 7068 . . . . 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)
3229, 30, 30, 31mp3an 1421 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
3320, 21, 32fvmpt 6239 . . 3 (𝑊 ∈ V → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
342, 33syl 17 . 2 (𝑊𝑋 → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
351, 34syl5eq 2667 1 (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865  ·𝑖cip 15867  0gc0g 16021  ocvcocv 19923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-ocv 19926
This theorem is referenced by:  ocvval  19930  elocv  19931
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