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

Theorem ocv0 21097
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 4357 . . 3 βˆ… βŠ† 𝑉
2 ocvz.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
3 eqid 2733 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
4 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2733 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
6 ocvz.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
72, 3, 4, 5, 6ocvval 21087 . . 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 4471 . . . 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 3435 . . 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 2764 1 ( βŠ₯ β€˜βˆ…) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  βˆ…c0 4283  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141  Β·π‘–cip 17143  0gc0g 17326  ocvcocv 21080
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-ocv 21083
This theorem is referenced by:  ocvz  21098  css1  21110
  Copyright terms: Public domain W3C validator