Theorem riotaocN 36339

Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 11614 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 2977	. . 3 ⊢ Ⅎ𝑦 ⊥
2		nfriota1 7115	. . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓)
3	1, 2	nffv 6674	. 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))
4		riotaoc.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
5		riotaoc.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
6	4, 5	opoccl 36324	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵)
7	4, 5	opoccl 36324	. 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵)
8		riotaoc.a	. 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓))
9		fveq2 6664	. 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))
10	4, 5	opoccl 36324	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵)
11	4, 5	opcon2b 36327	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥)))
12	10, 11	reuhypd 5311	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦))
13	3, 6, 7, 8, 9, 12	riotaxfrd 7142	1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃!wreu 3140  ‘cfv 6349  ℩crio 7107  Basecbs 16477  occoc 16567  OPcops 36302

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-dm 5559  df-iota 6308  df-fv 6357  df-riota 7108  df-ov 7153  df-oposet 36306

This theorem is referenced by:  glbconN  36507