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Theorem riotaocN 38545
Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12200 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
riotaoc.b 𝐵 = (Base‘𝐾)
riotaoc.o = (oc‘𝐾)
riotaoc.a (𝑥 = ( 𝑦) → (𝜑𝜓))
Assertion
Ref Expression
riotaocN ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥, ,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem riotaocN
StepHypRef Expression
1 nfcv 2902 . . 3 𝑦
2 nfriota1 7375 . . 3 𝑦(𝑦𝐵 𝜓)
31, 2nffv 6901 . 2 𝑦( ‘(𝑦𝐵 𝜓))
4 riotaoc.b . . 3 𝐵 = (Base‘𝐾)
5 riotaoc.o . . 3 = (oc‘𝐾)
64, 5opoccl 38530 . 2 ((𝐾 ∈ OP ∧ 𝑦𝐵) → ( 𝑦) ∈ 𝐵)
74, 5opoccl 38530 . 2 ((𝐾 ∈ OP ∧ (𝑦𝐵 𝜓) ∈ 𝐵) → ( ‘(𝑦𝐵 𝜓)) ∈ 𝐵)
8 riotaoc.a . 2 (𝑥 = ( 𝑦) → (𝜑𝜓))
9 fveq2 6891 . 2 (𝑦 = (𝑦𝐵 𝜓) → ( 𝑦) = ( ‘(𝑦𝐵 𝜓)))
104, 5opoccl 38530 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ( 𝑥) ∈ 𝐵)
114, 5opcon2b 38533 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵𝑦𝐵) → (𝑥 = ( 𝑦) ↔ 𝑦 = ( 𝑥)))
1210, 11reuhypd 5417 . 2 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = ( 𝑦))
133, 6, 7, 8, 9, 12riotaxfrd 7403 1 ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  ∃!wreu 3373  cfv 6543  crio 7367  Basecbs 17151  occoc 17212  OPcops 38508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-dm 5686  df-iota 6495  df-fv 6551  df-riota 7368  df-ov 7415  df-oposet 38512
This theorem is referenced by:  glbconN  38713  glbconNOLD  38714
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