Theorem ltrnset 39037

Description: The set of lattice translations for a fiducial co-atom 𝑊. (Contributed by NM, 11-May-2012.)

Hypotheses

Ref	Expression
ltrnset.l	⊢ ≤ = (le‘𝐾)
ltrnset.j	⊢ ∨ = (join‘𝐾)
ltrnset.m	⊢ ∧ = (meet‘𝐾)
ltrnset.a	⊢ 𝐴 = (Atoms‘𝐾)
ltrnset.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrnset.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
ltrnset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrnset	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})

Distinct variable groups:   𝑞,𝑝,𝐴   𝐷,𝑓   𝑓,𝑝,𝑞,𝐾   𝑓,𝑊,𝑝,𝑞

Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓,𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑓,𝑞,𝑝)   𝐻(𝑓,𝑞,𝑝)   ∨ (𝑓,𝑞,𝑝)   ≤ (𝑓,𝑞,𝑝)   ∧ (𝑓,𝑞,𝑝)

Proof of Theorem ltrnset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrnset.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
2		ltrnset.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		ltrnset.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		ltrnset.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
5		ltrnset.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		ltrnset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7	2, 3, 4, 5, 6	ltrnfset 39036	. . . 4 ⊢ (𝐾 ∈ 𝐵 → (LTrn‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}))
8	7	fveq1d 6894	. . 3 ⊢ (𝐾 ∈ 𝐵 → ((LTrn‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})‘𝑊))
9	1, 8	eqtrid 2785	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝑇 = ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})‘𝑊))
10		fveq2 6892	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = ((LDil‘𝐾)‘𝑊))
11		ltrnset.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2791	. . . 4 ⊢ (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = 𝐷)
13		breq2 5153	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑝 ≤ 𝑤 ↔ 𝑝 ≤ 𝑊))
14	13	notbid 318	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ 𝑝 ≤ 𝑤 ↔ ¬ 𝑝 ≤ 𝑊))
15		breq2 5153	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊))
16	15	notbid 318	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊))
17	14, 16	anbi12d 632	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
18		oveq2 7417	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊))
19		oveq2 7417	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))
20	18, 19	eqeq12d 2749	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤) ↔ ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊)))
21	17, 20	imbi12d 345	. . . . 5 ⊢ (𝑤 = 𝑊 → (((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤)) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))))
22	21	2ralbidv 3219	. . . 4 ⊢ (𝑤 = 𝑊 → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))))
23	12, 22	rabeqbidv 3450	. . 3 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))} = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})
24		eqid 2733	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))}) = (𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})
25	11	fvexi 6906	. . . 4 ⊢ 𝐷 ∈ V
26	25	rabex 5333	. . 3 ⊢ {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))} ∈ V
27	23, 24, 26	fvmpt 6999	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑤 ∧ ¬ 𝑞 ≤ 𝑤) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑤) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑤))})‘𝑊) = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})
28	9, 27	sylan9eq 2793	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → 𝑇 = {𝑓 ∈ 𝐷 ∣ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝑓‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝑓‘𝑞)) ∧ 𝑊))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433   class class class wbr 5149   ↦ cmpt 5232  ‘cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  meetcmee 18265  Atomscatm 38181  LHypclh 38903  LDilcldil 39019  LTrncltrn 39020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-ltrn 39024

This theorem is referenced by:  isltrn  39038