Theorem riotaocN 37150

Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 11884 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 2906	. . 3 ⊢ Ⅎ𝑦 ⊥
2		nfriota1 7219	. . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓)
3	1, 2	nffv 6766	. 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))
4		riotaoc.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
5		riotaoc.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
6	4, 5	opoccl 37135	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵)
7	4, 5	opoccl 37135	. 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵)
8		riotaoc.a	. 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓))
9		fveq2 6756	. 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))
10	4, 5	opoccl 37135	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵)
11	4, 5	opcon2b 37138	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥)))
12	10, 11	reuhypd 5337	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦))
13	3, 6, 7, 8, 9, 12	riotaxfrd 7247	1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃!wreu 3065  ‘cfv 6418  ℩crio 7211  Basecbs 16840  occoc 16896  OPcops 37113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-dm 5590  df-iota 6376  df-fv 6426  df-riota 7212  df-ov 7258  df-oposet 37117

This theorem is referenced by:  glbconN  37318