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

Theorem ocvfval 21093
Description: The orthocomplement operation. (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
ocvfval (π‘Š ∈ 𝑋 β†’ βŠ₯ = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
Distinct variable groups:   π‘₯,𝑠,𝑦, 0   𝑉,𝑠,π‘₯,𝑦   π‘Š,𝑠,π‘₯,𝑦   , ,𝑠,π‘₯,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦,𝑠)   βŠ₯ (π‘₯,𝑦,𝑠)   𝑋(π‘₯,𝑦,𝑠)

Proof of Theorem ocvfval
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 ocvfval.o . 2 βŠ₯ = (ocvβ€˜π‘Š)
2 elex 3465 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6846 . . . . . . 7 (β„Ž = π‘Š β†’ (Baseβ€˜β„Ž) = (Baseβ€˜π‘Š))
4 ocvfval.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2791 . . . . . 6 (β„Ž = π‘Š β†’ (Baseβ€˜β„Ž) = 𝑉)
65pweqd 4581 . . . . 5 (β„Ž = π‘Š β†’ 𝒫 (Baseβ€˜β„Ž) = 𝒫 𝑉)
7 fveq2 6846 . . . . . . . . . 10 (β„Ž = π‘Š β†’ (Β·π‘–β€˜β„Ž) = (Β·π‘–β€˜π‘Š))
8 ocvfval.i . . . . . . . . . 10 , = (Β·π‘–β€˜π‘Š)
97, 8eqtr4di 2791 . . . . . . . . 9 (β„Ž = π‘Š β†’ (Β·π‘–β€˜β„Ž) = , )
109oveqd 7378 . . . . . . . 8 (β„Ž = π‘Š β†’ (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = (π‘₯ , 𝑦))
11 fveq2 6846 . . . . . . . . . . 11 (β„Ž = π‘Š β†’ (Scalarβ€˜β„Ž) = (Scalarβ€˜π‘Š))
12 ocvfval.f . . . . . . . . . . 11 𝐹 = (Scalarβ€˜π‘Š)
1311, 12eqtr4di 2791 . . . . . . . . . 10 (β„Ž = π‘Š β†’ (Scalarβ€˜β„Ž) = 𝐹)
1413fveq2d 6850 . . . . . . . . 9 (β„Ž = π‘Š β†’ (0gβ€˜(Scalarβ€˜β„Ž)) = (0gβ€˜πΉ))
15 ocvfval.z . . . . . . . . 9 0 = (0gβ€˜πΉ)
1614, 15eqtr4di 2791 . . . . . . . 8 (β„Ž = π‘Š β†’ (0gβ€˜(Scalarβ€˜β„Ž)) = 0 )
1710, 16eqeq12d 2749 . . . . . . 7 (β„Ž = π‘Š β†’ ((π‘₯(Β·π‘–β€˜β„Ž)𝑦) = (0gβ€˜(Scalarβ€˜β„Ž)) ↔ (π‘₯ , 𝑦) = 0 ))
1817ralbidv 3171 . . . . . 6 (β„Ž = π‘Š β†’ (βˆ€π‘¦ ∈ 𝑠 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = (0gβ€˜(Scalarβ€˜β„Ž)) ↔ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 ))
195, 18rabeqbidv 3423 . . . . 5 (β„Ž = π‘Š β†’ {π‘₯ ∈ (Baseβ€˜β„Ž) ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = (0gβ€˜(Scalarβ€˜β„Ž))} = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })
206, 19mpteq12dv 5200 . . . 4 (β„Ž = π‘Š β†’ (𝑠 ∈ 𝒫 (Baseβ€˜β„Ž) ↦ {π‘₯ ∈ (Baseβ€˜β„Ž) ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = (0gβ€˜(Scalarβ€˜β„Ž))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
21 df-ocv 21090 . . . 4 ocv = (β„Ž ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜β„Ž) ↦ {π‘₯ ∈ (Baseβ€˜β„Ž) ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯(Β·π‘–β€˜β„Ž)𝑦) = (0gβ€˜(Scalarβ€˜β„Ž))}))
22 eqid 2733 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 })
234fvexi 6860 . . . . . . . 8 𝑉 ∈ V
24 ssrab2 4041 . . . . . . . 8 {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 } βŠ† 𝑉
2523, 24elpwi2 5307 . . . . . . 7 {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 } ∈ 𝒫 𝑉
2625a1i 11 . . . . . 6 (𝑠 ∈ 𝒫 𝑉 β†’ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 } ∈ 𝒫 𝑉)
2722, 26fmpti 7064 . . . . 5 (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }):𝒫 π‘‰βŸΆπ’« 𝑉
2823pwex 5339 . . . . 5 𝒫 𝑉 ∈ V
29 fex2 7874 . . . . 5 (((𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }):𝒫 π‘‰βŸΆπ’« 𝑉 ∧ 𝒫 𝑉 ∈ V ∧ 𝒫 𝑉 ∈ V) β†’ (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }) ∈ V)
3027, 28, 28, 29mp3an 1462 . . . 4 (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }) ∈ V
3120, 21, 30fvmpt 6952 . . 3 (π‘Š ∈ V β†’ (ocvβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
322, 31syl 17 . 2 (π‘Š ∈ 𝑋 β†’ (ocvβ€˜π‘Š) = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
331, 32eqtrid 2785 1 (π‘Š ∈ 𝑋 β†’ βŠ₯ = (𝑠 ∈ 𝒫 𝑉 ↦ {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ 𝑠 (π‘₯ , 𝑦) = 0 }))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3447  π’« cpw 4564   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  Scalarcsca 17144  Β·π‘–cip 17146  0gc0g 17329  ocvcocv 21087
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 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-ocv 21090
This theorem is referenced by:  ocvval  21094  elocv  21095
  Copyright terms: Public domain W3C validator