Theorem trlval 41187

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 41186	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)))))
10	9	fveq1d 6879	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑅‘𝐹) = ((𝑓 ∈ 𝑇 ↦ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))))‘𝐹))
11		fveq1 6876	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝))
12	11	oveq2d 7428	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑝 ∨ (𝑓‘𝑝)) = (𝑝 ∨ (𝐹‘𝑝)))
13	12	oveq1d 7427	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))
14	13	eqeq2d 2772	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) ↔ 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊)))
15	14	imbi2d 343	. . . . 5 ⊢ (𝑓 = 𝐹 → ((¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))
16	15	ralbidv 3186	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊)) ↔ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))))
17	16	riotabidv 7371	. . 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 2761	. . 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 7373	. . 3 ⊢ (℩𝑥 ∈ 𝐵 ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → 𝑥 = ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊))) ∈ V
20	17, 18, 19	fvmpt 6985	. 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 2816	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩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 3077   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6531  ℩crio 7368  (class class class)co 7412  Basecbs 17367  lecple 17415  joincjn 18465  meetcmee 18466  Atomscatm 40288  LHypclh 41009  LTrncltrn 41126  trLctrl 41183

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-trl 41184

This theorem is used by:  trlval2  41188