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Theorem ocv0 21713
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 4406 . . 3 ∅ ⊆ 𝑉
2 ocvz.v . . . 4 𝑉 = (Base‘𝑊)
3 eqid 2735 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2735 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2735 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 ocvz.o . . . 4 = (ocv‘𝑊)
72, 3, 4, 5, 6ocvval 21703 . . 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 4519 . . . 4 𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
109rgenw 3063 . . 3 𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
11 rabid2 3468 . . 3 (𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
1210, 11mpbir 231 . 2 𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
138, 12eqtr4i 2766 1 ( ‘∅) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wral 3059  {crab 3433  wss 3963  c0 4339  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  ocvcocv 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-ocv 21699
This theorem is referenced by:  ocvz  21714  css1  21726
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