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Theorem riotaocN 38591
Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12194 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 2897 . . 3 𝑦
2 nfriota1 7367 . . 3 𝑦(𝑦𝐵 𝜓)
31, 2nffv 6894 . 2 𝑦( ‘(𝑦𝐵 𝜓))
4 riotaoc.b . . 3 𝐵 = (Base‘𝐾)
5 riotaoc.o . . 3 = (oc‘𝐾)
64, 5opoccl 38576 . 2 ((𝐾 ∈ OP ∧ 𝑦𝐵) → ( 𝑦) ∈ 𝐵)
74, 5opoccl 38576 . 2 ((𝐾 ∈ OP ∧ (𝑦𝐵 𝜓) ∈ 𝐵) → ( ‘(𝑦𝐵 𝜓)) ∈ 𝐵)
8 riotaoc.a . 2 (𝑥 = ( 𝑦) → (𝜑𝜓))
9 fveq2 6884 . 2 (𝑦 = (𝑦𝐵 𝜓) → ( 𝑦) = ( ‘(𝑦𝐵 𝜓)))
104, 5opoccl 38576 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ( 𝑥) ∈ 𝐵)
114, 5opcon2b 38579 . . 3 ((𝐾 ∈ OP ∧ 𝑥𝐵𝑦𝐵) → (𝑥 = ( 𝑦) ↔ 𝑦 = ( 𝑥)))
1210, 11reuhypd 5410 . 2 ((𝐾 ∈ OP ∧ 𝑥𝐵) → ∃!𝑦𝐵 𝑥 = ( 𝑦))
133, 6, 7, 8, 9, 12riotaxfrd 7395 1 ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  ∃!wreu 3368  cfv 6536  crio 7359  Basecbs 17150  occoc 17211  OPcops 38554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6488  df-fv 6544  df-riota 7360  df-ov 7407  df-oposet 38558
This theorem is referenced by:  glbconN  38759  glbconNOLD  38760
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