Theorem trlset 41034

Description: The set of traces of lattice translations for a fiducial co-atom 𝑊. (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)
trlset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlset.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
trlset	⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))

Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑓,𝑝,𝑥,𝐾   𝑇,𝑓   𝑓,𝑊,𝑝,𝑥

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

Proof of Theorem trlset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlset.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
2		trlset.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		trlset.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		trlset.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		trlset.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
6		trlset.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		trlset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8	2, 3, 4, 5, 6, 7	trlfset 41033	. . . 4 ⊢ (𝐾 ∈ 𝐶 → (trL‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))))
9	8	fveq1d 6880	. . 3 ⊢ (𝐾 ∈ 𝐶 → ((trL‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))‘𝑊))
10	1, 9	eqtrid 2807	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝑅 = ((𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))‘𝑊))
11		fveq2 6878	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12		breq2 5107	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊))
13	12	notbid 321	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊))
14		oveq2 7421	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))
15	14	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) ↔ 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))
16	13, 15	imbi12d 347	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
17	16	ralbidv 3185	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
18	17	riotabidv 7372	. . . . 5 ⊢ (𝑤 = 𝑊 → (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
19	11, 18	mpteq12dv 5192	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))
20		eqid 2760	. . . 4 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤))))) = (𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))
21		fvex 6891	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
22	21	mptex 7222	. . . 4 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) ∈ V
23	19, 20, 22	fvmpt 6986	. . 3 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))‘𝑊) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))
24		trlset.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
25	24	mpteq1i 5196	. . 3 ⊢ (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
26	23, 25	eqtr4di 2813	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))‘𝑊) = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))
27	10, 26	sylan9eq 2815	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  ℩crio 7369  (class class class)co 7413  Basecbs 17301  lecple 17349  joincjn 18399  meetcmee 18400  Atomscatm 40136  LHypclh 40857  LTrncltrn 40974  trLctrl 41031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-trl 41032

This theorem is used by:  trlval  41035