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Theorem ocv0 19940
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 3944 . . 3 ∅ ⊆ 𝑉
2 ocvz.v . . . 4 𝑉 = (Base‘𝑊)
3 eqid 2621 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2621 . . . 4 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6 ocvz.o . . . 4 = (ocv‘𝑊)
72, 3, 4, 5, 6ocvval 19930 . . 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 4048 . . . 4 𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
109rgenw 2919 . . 3 𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))
11 rabid2 3107 . . 3 (𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑥𝑉𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
1210, 11mpbir 221 . 2 𝑉 = {𝑥𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
138, 12eqtr4i 2646 1 ( ‘∅) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wral 2907  {crab 2911  wss 3555  c0 3891  cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865  ·𝑖cip 15867  0gc0g 16021  ocvcocv 19923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-ocv 19926
This theorem is referenced by:  ocvz  19941  css1  19953
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