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Theorem riotaocN 39191
Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12245 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 2903 . . 3 𝑦
2 nfriota1 7395 . . 3 𝑦(𝑦𝐵 𝜓)
31, 2nffv 6917 . 2 𝑦( ‘(𝑦𝐵 𝜓))
4 riotaoc.b . . 3 𝐵 = (Base‘𝐾)
5 riotaoc.o . . 3 = (oc‘𝐾)
64, 5opoccl 39176 . 2 ((𝐾 ∈ OP ∧ 𝑦𝐵) → ( 𝑦) ∈ 𝐵)
74, 5opoccl 39176 . 2 ((𝐾 ∈ OP ∧ (𝑦𝐵 𝜓) ∈ 𝐵) → ( ‘(𝑦𝐵 𝜓)) ∈ 𝐵)
8 riotaoc.a . 2 (𝑥 = ( 𝑦) → (𝜑𝜓))
9 fveq2 6907 . 2 (𝑦 = (𝑦𝐵 𝜓) → ( 𝑦) = ( ‘(𝑦𝐵 𝜓)))
104, 5opoccl 39176 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ( 𝑥) ∈ 𝐵)
114, 5opcon2b 39179 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵𝑦𝐵) → (𝑥 = ( 𝑦) ↔ 𝑦 = ( 𝑥)))
1210, 11reuhypd 5425 . 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 1537  wcel 2106  ∃!wreu 3376  cfv 6563  crio 7387  Basecbs 17245  occoc 17306  OPcops 39154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-riota 7388  df-ov 7434  df-oposet 39158
This theorem is referenced by:  glbconN  39359  glbconNOLD  39360
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