Theorem trlset 40417

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 40416	. . . 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 6836	. . 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 2783	. 2 ⊢ (𝐾 ∈ 𝐶 → 𝑅 = ((𝑤 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)))))‘𝑊))
11		fveq2 6834	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12		breq2 5102	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊))
13	12	notbid 318	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊))
14		oveq2 7366	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))
15	14	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) ↔ 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))
16	13, 15	imbi12d 344	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
17	16	ralbidv 3159	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
18	17	riotabidv 7317	. . . . 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 5185	. . . 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 2736	. . . 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 6847	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
22	21	mptex 7169	. . . 4 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) ∈ V
23	19, 20, 22	fvmpt 6941	. . 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 5189	. . 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 2789	. 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 2791	1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098   ↦ cmpt 5179  ‘cfv 6492  ℩crio 7314  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  meetcmee 18235  Atomscatm 39519  LHypclh 40240  LTrncltrn 40357  trLctrl 40414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-trl 40415

This theorem is referenced by:  trlval  40418