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Theorem ocvfval 20133
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 3316 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6304 . . . . . . 7 ( = 𝑊 → (Base‘) = (Base‘𝑊))
4 ocvfval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2776 . . . . . 6 ( = 𝑊 → (Base‘) = 𝑉)
65pweqd 4271 . . . . 5 ( = 𝑊 → 𝒫 (Base‘) = 𝒫 𝑉)
7 fveq2 6304 . . . . . . . . . 10 ( = 𝑊 → (·𝑖) = (·𝑖𝑊))
8 ocvfval.i . . . . . . . . . 10 , = (·𝑖𝑊)
97, 8syl6eqr 2776 . . . . . . . . 9 ( = 𝑊 → (·𝑖) = , )
109oveqd 6782 . . . . . . . 8 ( = 𝑊 → (𝑥(·𝑖)𝑦) = (𝑥 , 𝑦))
11 fveq2 6304 . . . . . . . . . . 11 ( = 𝑊 → (Scalar‘) = (Scalar‘𝑊))
12 ocvfval.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1311, 12syl6eqr 2776 . . . . . . . . . 10 ( = 𝑊 → (Scalar‘) = 𝐹)
1413fveq2d 6308 . . . . . . . . 9 ( = 𝑊 → (0g‘(Scalar‘)) = (0g𝐹))
15 ocvfval.z . . . . . . . . 9 0 = (0g𝐹)
1614, 15syl6eqr 2776 . . . . . . . 8 ( = 𝑊 → (0g‘(Scalar‘)) = 0 )
1710, 16eqeq12d 2739 . . . . . . 7 ( = 𝑊 → ((𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ (𝑥 , 𝑦) = 0 ))
1817ralbidv 3088 . . . . . 6 ( = 𝑊 → (∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘)) ↔ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 ))
195, 18rabeqbidv 3299 . . . . 5 ( = 𝑊 → {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))} = {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
206, 19mpteq12dv 4841 . . . 4 ( = 𝑊 → (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
21 df-ocv 20130 . . . 4 ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))
22 eqid 2724 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 })
23 ssrab2 3793 . . . . . . . 8 {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
24 fvex 6314 . . . . . . . . . 10 (Base‘𝑊) ∈ V
254, 24eqeltri 2799 . . . . . . . . 9 𝑉 ∈ V
2625elpw2 4933 . . . . . . . 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 6498 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
3025pwex 4953 . . . . 5 𝒫 𝑉 ∈ V
31 fex2 7238 . . . . 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 1537 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
3320, 21, 32fvmpt 6396 . . 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 2770 1 (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  wral 3014  {crab 3018  Vcvv 3304  wss 3680  𝒫 cpw 4266  cmpt 4837  wf 5997  cfv 6001  (class class class)co 6765  Basecbs 15980  Scalarcsca 16067  ·𝑖cip 16069  0gc0g 16223  ocvcocv 20127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-ocv 20130
This theorem is referenced by:  ocvval  20134  elocv  20135
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