Theorem ltrnset 40107

Description: The set of lattice translations for a fiducial co-atom 𝑊. (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)
ltrnset.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
ltrnset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrnset	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})

Distinct variable groups:   𝑞,𝑝,𝐴   𝐷,𝑓   𝑓,𝑝,𝑞,𝐾   𝑓,𝑊,𝑝,𝑞

Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓,𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑓,𝑞,𝑝)   𝐻(𝑓,𝑞,𝑝)   ∨ (𝑓,𝑞,𝑝)   ≤ (𝑓,𝑞,𝑝)   ∧ (𝑓,𝑞,𝑝)

Proof of Theorem ltrnset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrnset.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
2		ltrnset.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		ltrnset.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		ltrnset.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
5		ltrnset.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		ltrnset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7	2, 3, 4, 5, 6	ltrnfset 40106	. . . 4 ⊢ (𝐾 ∈ 𝐵 → (LTrn‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}))
8	7	fveq1d 6862	. . 3 ⊢ (𝐾 ∈ 𝐵 → ((LTrn‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})‘𝑊))
9	1, 8	eqtrid 2777	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝑇 = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})‘𝑊))
10		fveq2 6860	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = ((LDil‘𝐾)‘𝑊))
11		ltrnset.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2783	. . . 4 ⊢ (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = 𝐷)
13		breq2 5113	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊))
14	13	notbid 318	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊))
15		breq2 5113	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊))
16	15	notbid 318	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊))
17	14, 16	anbi12d 632	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
18		oveq2 7397	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))
19		oveq2 7397	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))
20	18, 19	eqeq12d 2746	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤) ↔ ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊)))
21	17, 20	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑊 → (((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤)) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))))
22	21	2ralbidv 3202	. . . 4 ⊢ (𝑤 = 𝑊 → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))))
23	12, 22	rabeqbidv 3427	. . 3 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))} = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})
24		eqid 2730	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})
25	11	fvexi 6874	. . . 4 ⊢ 𝐷 ∈ V
26	25	rabex 5296	. . 3 ⊢ {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))} ∈ V
27	23, 24, 26	fvmpt 6970	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})‘𝑊) = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})
28	9, 27	sylan9eq 2785	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝐷 ∣ ∀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 3045  {crab 3408   class class class wbr 5109   ↦ cmpt 5190  ‘cfv 6513  (class class class)co 7389  lecple 17233  joincjn 18278  meetcmee 18279  Atomscatm 39251  LHypclh 39973  LDilcldil 40089  LTrncltrn 40090

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-ltrn 40094

This theorem is referenced by:  isltrn  40108