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Theorem ocv0 21538
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 4388 . . 3 ∅ ⊆ 𝑉
2 ocvz.v . . . 4 𝑉 = (Base‘𝑊)
3 eqid 2724 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2724 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2724 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 ocvz.o . . . 4 = (ocv‘𝑊)
72, 3, 4, 5, 6ocvval 21528 . . 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 4504 . . . 4 𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
109rgenw 3057 . . 3 𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
11 rabid2 3456 . . 3 (𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
1210, 11mpbir 230 . 2 𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
138, 12eqtr4i 2755 1 ( ‘∅) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wral 3053  {crab 3424  wss 3940  c0 4314  cfv 6533  (class class class)co 7401  Basecbs 17143  Scalarcsca 17199  ·𝑖cip 17201  0gc0g 17384  ocvcocv 21521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-ocv 21524
This theorem is referenced by:  ocvz  21539  css1  21551
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