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Theorem riotaocN 35011
Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 11203 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 2912 . . 3 𝑦
2 nfriota1 6760 . . 3 𝑦(𝑦𝐵 𝜓)
31, 2nffv 6339 . 2 𝑦( ‘(𝑦𝐵 𝜓))
4 riotaoc.b . . 3 𝐵 = (Base‘𝐾)
5 riotaoc.o . . 3 = (oc‘𝐾)
64, 5opoccl 34996 . 2 ((𝐾 ∈ OP ∧ 𝑦𝐵) → ( 𝑦) ∈ 𝐵)
74, 5opoccl 34996 . 2 ((𝐾 ∈ OP ∧ (𝑦𝐵 𝜓) ∈ 𝐵) → ( ‘(𝑦𝐵 𝜓)) ∈ 𝐵)
8 riotaoc.a . 2 (𝑥 = ( 𝑦) → (𝜑𝜓))
9 fveq2 6332 . 2 (𝑦 = (𝑦𝐵 𝜓) → ( 𝑦) = ( ‘(𝑦𝐵 𝜓)))
104, 5opoccl 34996 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ( 𝑥) ∈ 𝐵)
114, 5opcon2b 34999 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵𝑦𝐵) → (𝑥 = ( 𝑦) ↔ 𝑦 = ( 𝑥)))
1210, 11reuhypd 5023 . 2 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = ( 𝑦))
133, 6, 7, 8, 9, 12riotaxfrd 6784 1 ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  ∃!wreu 3062  cfv 6031  crio 6752  Basecbs 16063  occoc 16156  OPcops 34974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-dm 5259  df-iota 5994  df-fv 6039  df-riota 6753  df-ov 6795  df-oposet 34978
This theorem is referenced by:  glbconN  35178
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