Theorem ltrnfset 40111

Description: The set of all lattice translations for a lattice 𝐾. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
ltrnset.l	⊢ ≤ = (le‘𝐾)
ltrnset.j	⊢ ∨ = (join‘𝐾)
ltrnset.m	⊢ ∧ = (meet‘𝐾)
ltrnset.a	⊢ 𝐴 = (Atoms‘𝐾)
ltrnset.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
ltrnfset	⊢ (𝐾 ∈ 𝐶 → (LTrn‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}))

Distinct variable groups:   𝑞,𝑝,𝐴   𝑤,𝐻   𝑓,𝑝,𝑞,𝑤,𝐾

Allowed substitution hints:   𝐴(𝑤,𝑓)   𝐶(𝑤,𝑓,𝑞,𝑝)   𝐻(𝑓,𝑞,𝑝)   ∨ (𝑤,𝑓,𝑞,𝑝)   ≤ (𝑤,𝑓,𝑞,𝑝)   ∧ (𝑤,𝑓,𝑞,𝑝)

Proof of Theorem ltrnfset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3468	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		fveq2 6858	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		ltrnset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2782	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6858	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LDil‘𝑘) = (LDil‘𝐾))
6	5	fveq1d 6860	. . . . 5 ⊢ (𝑘 = 𝐾 → ((LDil‘𝑘)‘𝑤) = ((LDil‘𝐾)‘𝑤))
7		fveq2 6858	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
8		ltrnset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
9	7, 8	eqtr4di 2782	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
10		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
11		ltrnset.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
12	10, 11	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
13	12	breqd 5118	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤 ↔ 𝑝 ≤ 𝑤))
14	13	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 ≤ 𝑤))
15	12	breqd 5118	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤 ↔ 𝑞 ≤ 𝑤))
16	15	notbid 318	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 ≤ 𝑤))
17	14, 16	anbi12d 632	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) ↔ (¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤)))
18		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19		ltrnset.m	. . . . . . . . . . 11 ⊢ ∧ = (meet‘𝐾)
20	18, 19	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
21		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22		ltrnset.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
23	21, 22	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
24	23	oveqd 7404	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓‘𝑝)) = (𝑝 ∨ (𝑓‘𝑝)))
25		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤)
26	20, 24, 25	oveq123d 7408	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))
27	23	oveqd 7404	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑓‘𝑞)) = (𝑞 ∨ (𝑓‘𝑞)))
28	20, 27, 25	oveq123d 7408	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))
29	26, 28	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤) ↔ ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤)))
30	17, 29	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤)) ↔ ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))))
31	9, 30	raleqbidv 3319	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤)) ↔ ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))))
32	9, 31	raleqbidv 3319	. . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))))
33	6, 32	rabeqbidv 3424	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤))} = {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})
34	4, 33	mpteq12dv 5194	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤))}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}))
35		df-ltrn 40099	. . 3 ⊢ LTrn = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤))}))
36	34, 35, 3	mptfvmpt 7202	. 2 ⊢ (𝐾 ∈ V → (LTrn‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}))
37	1, 36	syl 17	1 ⊢ (𝐾 ∈ 𝐶 → (LTrn‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   class class class wbr 5107   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  lecple 17227  joincjn 18272  meetcmee 18273  Atomscatm 39256  LHypclh 39978  LDilcldil 40094  LTrncltrn 40095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-ltrn 40099

This theorem is referenced by:  ltrnset  40112