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Theorem riotaocN 39210
Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12247 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 2905 . . 3 𝑦
2 nfriota1 7395 . . 3 𝑦(𝑦𝐵 𝜓)
31, 2nffv 6916 . 2 𝑦( ‘(𝑦𝐵 𝜓))
4 riotaoc.b . . 3 𝐵 = (Base‘𝐾)
5 riotaoc.o . . 3 = (oc‘𝐾)
64, 5opoccl 39195 . 2 ((𝐾 ∈ OP ∧ 𝑦𝐵) → ( 𝑦) ∈ 𝐵)
74, 5opoccl 39195 . 2 ((𝐾 ∈ OP ∧ (𝑦𝐵 𝜓) ∈ 𝐵) → ( ‘(𝑦𝐵 𝜓)) ∈ 𝐵)
8 riotaoc.a . 2 (𝑥 = ( 𝑦) → (𝜑𝜓))
9 fveq2 6906 . 2 (𝑦 = (𝑦𝐵 𝜓) → ( 𝑦) = ( ‘(𝑦𝐵 𝜓)))
104, 5opoccl 39195 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ( 𝑥) ∈ 𝐵)
114, 5opcon2b 39198 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵𝑦𝐵) → (𝑥 = ( 𝑦) ↔ 𝑦 = ( 𝑥)))
1210, 11reuhypd 5419 . 2 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = ( 𝑦))
133, 6, 7, 8, 9, 12riotaxfrd 7422 1 ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!wreu 3378  cfv 6561  crio 7387  Basecbs 17247  occoc 17305  OPcops 39173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-riota 7388  df-ov 7434  df-oposet 39177
This theorem is referenced by:  glbconN  39378  glbconNOLD  39379
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