Theorem trlset 40660

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 40659	. . . 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 2787	. 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 5083	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊))
13	12	notbid 319	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊))
14		oveq2 7371	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))
15	14	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) ↔ 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))
16	13, 15	imbi12d 345	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
17	16	ralbidv 3163	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑤 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
18	17	riotabidv 7322	. . . . 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 5166	. . . 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 2740	. . . 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 7174	. . . 4 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) ∈ V
23	19, 20, 22	fvmpt 6942	. . 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 5170	. . 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 2793	. 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 2795	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 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054   class class class wbr 5079   ↦ cmpt 5160  ‘cfv 6492  ℩crio 7319  (class class class)co 7363  Basecbs 17177  lecple 17225  joincjn 18275  meetcmee 18276  Atomscatm 39762  LHypclh 40483  LTrncltrn 40600  trLctrl 40657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7320  df-ov 7366  df-trl 40658

This theorem is referenced by:  trlval  40661