Theorem riotaocN 38713

Description: The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12231 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 2899	. . 3 ⊢ Ⅎ𝑦 ⊥
2		nfriota1 7389	. . 3 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐵 𝜓)
3	1, 2	nffv 6912	. 2 ⊢ Ⅎ𝑦( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓))
4		riotaoc.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
5		riotaoc.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
6	4, 5	opoccl 38698	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑦 ∈ 𝐵) → ( ⊥ ‘𝑦) ∈ 𝐵)
7	4, 5	opoccl 38698	. 2 ⊢ ((𝐾 ∈ OP ∧ (℩𝑦 ∈ 𝐵 𝜓) ∈ 𝐵) → ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)) ∈ 𝐵)
8		riotaoc.a	. 2 ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓))
9		fveq2 6902	. 2 ⊢ (𝑦 = (℩𝑦 ∈ 𝐵 𝜓) → ( ⊥ ‘𝑦) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))
10	4, 5	opoccl 38698	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ( ⊥ ‘𝑥) ∈ 𝐵)
11	4, 5	opcon2b 38701	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 = ( ⊥ ‘𝑦) ↔ 𝑦 = ( ⊥ ‘𝑥)))
12	10, 11	reuhypd 5423	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 𝑥 = ( ⊥ ‘𝑦))
13	3, 6, 7, 8, 9, 12	riotaxfrd 7417	1 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃!wreu 3372  ‘cfv 6553  ℩crio 7381  Basecbs 17187  occoc 17248  OPcops 38676

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 2166  ax-ext 2699  ax-nul 5310

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561  df-riota 7382  df-ov 7429  df-oposet 38680

This theorem is referenced by:  glbconN  38881  glbconNOLD  38882