MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocv0 Structured version   Visualization version   GIF version

Theorem ocv0 21229
Description: The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
ocvz.v 𝑉 = (Baseβ€˜π‘Š)
ocvz.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
ocv0 ( βŠ₯ β€˜βˆ…) = 𝑉

Proof of Theorem ocv0
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 4396 . . 3 βˆ… βŠ† 𝑉
2 ocvz.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
3 eqid 2732 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
4 eqid 2732 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2732 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
6 ocvz.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
72, 3, 4, 5, 6ocvval 21219 . . 3 (βˆ… βŠ† 𝑉 β†’ ( βŠ₯ β€˜βˆ…) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
81, 7ax-mp 5 . 2 ( βŠ₯ β€˜βˆ…) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}
9 ral0 4512 . . . 4 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))
109rgenw 3065 . . 3 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))
11 rabid2 3464 . . 3 (𝑉 = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ↔ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
1210, 11mpbir 230 . 2 𝑉 = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}
138, 12eqtr4i 2763 1 ( βŠ₯ β€˜βˆ…) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  βˆ…c0 4322  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  Scalarcsca 17199  Β·π‘–cip 17201  0gc0g 17384  ocvcocv 21212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-ocv 21215
This theorem is referenced by:  ocvz  21230  css1  21242
  Copyright terms: Public domain W3C validator