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Theorem ocv0 21540
Description: The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
ocvz.v 𝑉 = (Baseβ€˜π‘Š)
ocvz.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
ocv0 ( βŠ₯ β€˜βˆ…) = 𝑉

Proof of Theorem ocv0
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 4389 . . 3 βˆ… βŠ† 𝑉
2 ocvz.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
3 eqid 2724 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
4 eqid 2724 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2724 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
6 ocvz.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
72, 3, 4, 5, 6ocvval 21530 . . 3 (βˆ… βŠ† 𝑉 β†’ ( βŠ₯ β€˜βˆ…) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
81, 7ax-mp 5 . 2 ( βŠ₯ β€˜βˆ…) = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}
9 ral0 4505 . . . 4 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))
109rgenw 3057 . . 3 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))
11 rabid2 3456 . . 3 (𝑉 = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ↔ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
1210, 11mpbir 230 . 2 𝑉 = {π‘₯ ∈ 𝑉 ∣ βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}
138, 12eqtr4i 2755 1 ( βŠ₯ β€˜βˆ…) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  βˆ€wral 3053  {crab 3424   βŠ† wss 3941  βˆ…c0 4315  β€˜cfv 6534  (class class class)co 7402  Basecbs 17145  Scalarcsca 17201  Β·π‘–cip 17203  0gc0g 17386  ocvcocv 21523
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 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-ocv 21526
This theorem is referenced by:  ocvz  21541  css1  21553
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