Theorem trlfset 38101

Description: The set of all traces of lattice translations for a lattice 𝐾. (Contributed by NM, 20-May-2012.)

Hypotheses

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

Assertion

Ref	Expression
trlfset	⊢ (𝐾 ∈ 𝐶 → (trL‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))))

Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑤,𝐻   𝑓,𝑝,𝑤,𝑥,𝐾

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

Proof of Theorem trlfset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝐾 ∈ V)
2		fveq2 6756	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		trlset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2797	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6756	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6758	. . . . 5 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7		fveq2 6756	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
8		trlset.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
9	7, 8	eqtr4di 2797	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
10		fveq2 6756	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
11		trlset.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11	eqtr4di 2797	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
13		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14		trlset.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
15	13, 14	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
16	15	breqd 5081	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑤 ↔ 𝑝 ≤ 𝑤))
17	16	notbid 317	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑤 ↔ ¬ 𝑝 ≤ 𝑤))
18		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
19		trlset.m	. . . . . . . . . . 11 ⊢ ∧ = (meet‘𝐾)
20	18, 19	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
21		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
22		trlset.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
23	21, 22	eqtr4di 2797	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
24	23	oveqd 7272	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(join‘𝑘)(𝑓‘𝑝)) = (𝑝 ∨ (𝑓‘𝑝)))
25		eqidd 2739	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤)
26	20, 24, 25	oveq123d 7276	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))
27	26	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) ↔ 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))
28	17, 27	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤)) ↔ (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))
29	12, 28	raleqbidv 3327	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))
30	9, 29	riotaeqbidv 7215	. . . . 5 ⊢ (𝑘 = 𝐾 → (℩𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤))) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))
31	6, 30	mpteq12dv 5161	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (℩𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))
32	4, 31	mpteq12dv 5161	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (℩𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤))))) = (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))))
33		df-trl 38100	. . 3 ⊢ trL = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (℩𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤))))))
34	32, 33, 3	mptfvmpt 7086	. 2 ⊢ (𝐾 ∈ V → (trL‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))))
35	1, 34	syl 17	1 ⊢ (𝐾 ∈ 𝐶 → (trL‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  meetcmee 17945  Atomscatm 37204  LHypclh 37925  LTrncltrn 38042  trLctrl 38099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-trl 38100

This theorem is referenced by:  trlset  38102