Theorem riotaocN 36347

Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 11622 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) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
riotaocN	⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		nfcv 2979	. . 3 ⊢ Ⅎ𝑦 ⊥
2		nfriota1 7123	. . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓)
3	1, 2	nffv 6682	. 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))
4		riotaoc.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
5		riotaoc.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
6	4, 5	opoccl 36332	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵)
7	4, 5	opoccl 36332	. 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵)
8		riotaoc.a	. 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓))
9		fveq2 6672	. 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))
10	4, 5	opoccl 36332	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵)
11	4, 5	opcon2b 36335	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥)))
12	10, 11	reuhypd 5322	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦))
13	3, 6, 7, 8, 9, 12	riotaxfrd 7150	1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃!wreu 3142  ‘cfv 6357  ℩crio 7115  Basecbs 16485  occoc 16575  OPcops 36310

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-dm 5567  df-iota 6316  df-fv 6365  df-riota 7116  df-ov 7161  df-oposet 36314

This theorem is referenced by:  glbconN  36515