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Theorem ocv0 21695
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 4400 . . 3 ∅ ⊆ 𝑉
2 ocvz.v . . . 4 𝑉 = (Base‘𝑊)
3 eqid 2737 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2737 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2737 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 ocvz.o . . . 4 = (ocv‘𝑊)
72, 3, 4, 5, 6ocvval 21685 . . 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 4513 . . . 4 𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
109rgenw 3065 . . 3 𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
11 rabid2 3470 . . 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 2768 1 ( ‘∅) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wral 3061  {crab 3436  wss 3951  c0 4333  cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300  ·𝑖cip 17302  0gc0g 17484  ocvcocv 21678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-ocv 21681
This theorem is referenced by:  ocvz  21696  css1  21708
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