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Theorem ocv0 21603
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 4393 . . 3 βˆ… βŠ† 𝑉
2 ocvz.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
3 eqid 2728 . . . 4 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
4 eqid 2728 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5 eqid 2728 . . . 4 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
6 ocvz.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
72, 3, 4, 5, 6ocvval 21593 . . 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 4509 . . . 4 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))
109rgenw 3061 . . 3 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ βˆ… (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))
11 rabid2 3460 . . 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 2759 1 ( βŠ₯ β€˜βˆ…) = 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  βˆ€wral 3057  {crab 3428   βŠ† wss 3945  βˆ…c0 4319  β€˜cfv 6543  (class class class)co 7415  Basecbs 17174  Scalarcsca 17230  Β·π‘–cip 17232  0gc0g 17415  ocvcocv 21586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-ocv 21589
This theorem is referenced by:  ocvz  21604  css1  21616
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