Theorem ltrnldil 40079

Description: A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Hypotheses

Ref	Expression
ltrnldil.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrnldil.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
ltrnldil.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrnldil	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐷)

Proof of Theorem ltrnldil

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
2		eqid 2740	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2740	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
4		eqid 2740	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		ltrnldil.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
6		ltrnldil.d	. . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
7		ltrnldil.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	isltrn 40076	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹‘𝑞))(meet‘𝐾)𝑊)))))
9	8	simprbda 498	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  lecple 17318  joincjn 18381  meetcmee 18382  Atomscatm 39219  LHypclh 39941  LDilcldil 40057  LTrncltrn 40058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-ltrn 40062

This theorem is referenced by:  ltrnlaut  40080  ltrnval1  40091  ltrncnv  40103  ltrnco  40676