Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isltrn Structured version   Visualization version   GIF version

Theorem isltrn 39766
Description: The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (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
isltrn ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
Distinct variable groups:   𝑞,𝑝,𝐴   𝐾,𝑝,𝑞   𝑊,𝑝,𝑞   𝐹,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑞,𝑝)   𝐻(𝑞,𝑝)   (𝑞,𝑝)   (𝑞,𝑝)   (𝑞,𝑝)

Proof of Theorem isltrn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ltrnset.l . . . 4 = (le‘𝐾)
2 ltrnset.j . . . 4 = (join‘𝐾)
3 ltrnset.m . . . 4 = (meet‘𝐾)
4 ltrnset.a . . . 4 𝐴 = (Atoms‘𝐾)
5 ltrnset.h . . . 4 𝐻 = (LHyp‘𝐾)
6 ltrnset.d . . . 4 𝐷 = ((LDil‘𝐾)‘𝑊)
7 ltrnset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7ltrnset 39765 . . 3 ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
98eleq2d 2811 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇𝐹 ∈ {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))}))
10 fveq1 6899 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1110oveq2d 7439 . . . . . . 7 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1211oveq1d 7438 . . . . . 6 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
13 fveq1 6899 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1413oveq2d 7439 . . . . . . 7 (𝑓 = 𝐹 → (𝑞 (𝑓𝑞)) = (𝑞 (𝐹𝑞)))
1514oveq1d 7438 . . . . . 6 (𝑓 = 𝐹 → ((𝑞 (𝑓𝑞)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
1612, 15eqeq12d 2741 . . . . 5 (𝑓 = 𝐹 → (((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊) ↔ ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
1716imbi2d 339 . . . 4 (𝑓 = 𝐹 → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
18172ralbidv 3208 . . 3 (𝑓 = 𝐹 → (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
1918elrab 3680 . 2 (𝐹 ∈ {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))} ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
209, 19bitrdi 286 1 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418   class class class wbr 5152  cfv 6553  (class class class)co 7423  lecple 17268  joincjn 18331  meetcmee 18332  Atomscatm 38909  LHypclh 39631  LDilcldil 39747  LTrncltrn 39748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-ltrn 39752
This theorem is referenced by:  isltrn2N  39767  ltrnu  39768  ltrnldil  39769  ltrncnv  39793  idltrn  39797  cdleme50ltrn  40204  ltrnco  40366
  Copyright terms: Public domain W3C validator