Theorem ltrnset 38654

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 38653	. . . 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 6849	. . 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 2783	. 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 6847	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = ((LDil‘𝐾)‘𝑊))
11		ltrnset.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2789	. . . 4 ⊢ (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = 𝐷)
13		breq2 5114	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊))
14	13	notbid 317	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊))
15		breq2 5114	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊))
16	15	notbid 317	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊))
17	14, 16	anbi12d 631	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
18		oveq2 7370	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))
19		oveq2 7370	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))
20	18, 19	eqeq12d 2747	. . . . . 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 3208	. . . 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 3422	. . 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 2731	. . 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 6861	. . . 4 ⊢ 𝐷 ∈ V
26	25	rabex 5294	. . 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 6953	. 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 2791	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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3405   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6501  (class class class)co 7362  lecple 17154  joincjn 18214  meetcmee 18215  Atomscatm 37798  LHypclh 38520  LDilcldil 38636  LTrncltrn 38637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  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 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-ltrn 38641

This theorem is referenced by:  isltrn  38655