Theorem trlval 40418

Description: The value of the trace of a lattice translation. (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
trlval	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))

Distinct variable groups:   𝐴,𝑝   𝑥,𝐵   𝑥,𝑝,𝐾   𝑊,𝑝,𝑥   𝐹,𝑝,𝑥

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

Proof of Theorem trlval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trlset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		trlset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		trlset.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		trlset.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		trlset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		trlset.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		trlset.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		trlset.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	trlset 40417	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))
10	9	fveq1d 6836	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑅‘𝐹) = ((𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))‘𝐹))
11		fveq1 6833	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝))
12	11	oveq2d 7374	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑝 ∨ (𝑓‘𝑝)) = (𝑝 ∨ (𝐹‘𝑝)))
13	12	oveq1d 7373	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))
14	13	eqeq2d 2747	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) ↔ 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))
15	14	imbi2d 340	. . . . 5 ⊢ (𝑓 = 𝐹 → ((¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))
16	15	ralbidv 3159	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))
17	16	riotabidv 7317	. . 3 ⊢ (𝑓 = 𝐹 → (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))
18		eqid 2736	. . 3 ⊢ (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))) = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))
19		riotaex 7319	. . 3 ⊢ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))) ∈ V
20	17, 18, 19	fvmpt 6941	. 2 ⊢ (𝐹 ∈ 𝑇 → ((𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))‘𝐹) = (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))
21	10, 20	sylan9eq 2791	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩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:  trlval2  40419