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Theorem ocv0 19840
Description: The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
ocvz.v 𝑉 = (Base‘𝑊)
ocvz.o = (ocv‘𝑊)
Assertion
Ref Expression
ocv0 ( ‘∅) = 𝑉

Proof of Theorem ocv0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3924 . . 3 ∅ ⊆ 𝑉
2 ocvz.v . . . 4 𝑉 = (Base‘𝑊)
3 eqid 2610 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2610 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 ocvz.o . . . 4 = (ocv‘𝑊)
72, 3, 4, 5, 6ocvval 19830 . . 3 (∅ ⊆ 𝑉 → ( ‘∅) = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
81, 7ax-mp 5 . 2 ( ‘∅) = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
9 ral0 4028 . . . 4 𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
109rgenw 2908 . . 3 𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
11 rabid2 3096 . . 3 (𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
1210, 11mpbir 220 . 2 𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
138, 12eqtr4i 2635 1 ( ‘∅) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wral 2896  {crab 2900  wss 3540  c0 3874  cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771  ·𝑖cip 15773  0gc0g 15923  ocvcocv 19823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-ocv 19826
This theorem is referenced by:  ocvz  19841  css1  19853
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