MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocvfval Structured version   Visualization version   GIF version

Theorem ocvfval 20227
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 3364 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6332 . . . . . . 7 ( = 𝑊 → (Base‘) = (Base‘𝑊))
4 ocvfval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2823 . . . . . 6 ( = 𝑊 → (Base‘) = 𝑉)
65pweqd 4302 . . . . 5 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
7 fveq2 6332 . . . . . . . . . 10 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
8 ocvfval.i . . . . . . . . . 10 , = (·𝑖𝑊)
97, 8syl6eqr 2823 . . . . . . . . 9 ( = 𝑊 → (·𝑖) = , )
109oveqd 6810 . . . . . . . 8 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
11 fveq2 6332 . . . . . . . . . . 11 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
12 ocvfval.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1311, 12syl6eqr 2823 . . . . . . . . . 10 ( = 𝑊 → (Scalar‘) = 𝐹)
1413fveq2d 6336 . . . . . . . . 9 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
15 ocvfval.z . . . . . . . . 9 0 = (0g𝐹)
1614, 15syl6eqr 2823 . . . . . . . 8 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
1710, 16eqeq12d 2786 . . . . . . 7 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ (𝑥 , 𝑦) = 0 ))
1817ralbidv 3135 . . . . . 6 ( = 𝑊 → (∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 ))
195, 18rabeqbidv 3345 . . . . 5 ( = 𝑊 → {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))} = {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
206, 19mpteq12dv 4867 . . . 4 ( = 𝑊 → (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
21 df-ocv 20224 . . . 4 ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))
22 eqid 2771 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
23 ssrab2 3836 . . . . . . . 8 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
24 fvex 6342 . . . . . . . . . 10 (Base‘𝑊) ∈ V
254, 24eqeltri 2846 . . . . . . . . 9 𝑉 ∈ V
2625elpw2 4959 . . . . . . . 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 6525 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
3025pwex 4981 . . . . 5 𝒫 𝑉 ∈ V
31 fex2 7268 . . . . 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 1572 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
3320, 21, 32fvmpt 6424 . . 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 2817 1 (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4297  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  Scalarcsca 16152  ·𝑖cip 16154  0gc0g 16308  ocvcocv 20221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-ocv 20224
This theorem is referenced by:  ocvval  20228  elocv  20229
  Copyright terms: Public domain W3C validator