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Theorem ltrnu 34425
 Description: Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom 𝑊. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnu.l = (le‘𝐾)
ltrnu.j = (join‘𝐾)
ltrnu.m = (meet‘𝐾)
ltrnu.a 𝐴 = (Atoms‘𝐾)
ltrnu.h 𝐻 = (LHyp‘𝐾)
ltrnu.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrnu ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))

Proof of Theorem ltrnu
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 an4 861 . . 3 (((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ↔ ((𝑃𝐴𝑄𝐴) ∧ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊)))
2 simpr 476 . . . . 5 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → (𝑃𝐴𝑄𝐴))
3 simplr 788 . . . . . 6 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → 𝐹𝑇)
4 ltrnu.l . . . . . . . . 9 = (le‘𝐾)
5 ltrnu.j . . . . . . . . 9 = (join‘𝐾)
6 ltrnu.m . . . . . . . . 9 = (meet‘𝐾)
7 ltrnu.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
8 ltrnu.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
9 eqid 2610 . . . . . . . . 9 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10 ltrnu.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10isltrn 34423 . . . . . . . 8 ((𝐾𝑉𝑊𝐻) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
1211ad2antrr 758 . . . . . . 7 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
13 simpr 476 . . . . . . 7 ((𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
1412, 13syl6bi 242 . . . . . 6 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → (𝐹𝑇 → ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
153, 14mpd 15 . . . . 5 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
16 breq1 4586 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 𝑊𝑃 𝑊))
1716notbid 307 . . . . . . . 8 (𝑝 = 𝑃 → (¬ 𝑝 𝑊 ↔ ¬ 𝑃 𝑊))
1817anbi1d 737 . . . . . . 7 (𝑝 = 𝑃 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊)))
19 id 22 . . . . . . . . . 10 (𝑝 = 𝑃𝑝 = 𝑃)
20 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
2119, 20oveq12d 6567 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 (𝐹𝑝)) = (𝑃 (𝐹𝑃)))
2221oveq1d 6564 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑃 (𝐹𝑃)) 𝑊))
2322eqeq1d 2612 . . . . . . 7 (𝑝 = 𝑃 → (((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) ↔ ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2418, 23imbi12d 333 . . . . . 6 (𝑝 = 𝑃 → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
25 breq1 4586 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑞 𝑊𝑄 𝑊))
2625notbid 307 . . . . . . . 8 (𝑞 = 𝑄 → (¬ 𝑞 𝑊 ↔ ¬ 𝑄 𝑊))
2726anbi2d 736 . . . . . . 7 (𝑞 = 𝑄 → ((¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊)))
28 id 22 . . . . . . . . . 10 (𝑞 = 𝑄𝑞 = 𝑄)
29 fveq2 6103 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
3028, 29oveq12d 6567 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑞 (𝐹𝑞)) = (𝑄 (𝐹𝑄)))
3130oveq1d 6564 . . . . . . . 8 (𝑞 = 𝑄 → ((𝑞 (𝐹𝑞)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
3231eqeq2d 2620 . . . . . . 7 (𝑞 = 𝑄 → (((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊) ↔ ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊)))
3327, 32imbi12d 333 . . . . . 6 (𝑞 = 𝑄 → (((¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))))
3424, 33rspc2v 3293 . . . . 5 ((𝑃𝐴𝑄𝐴) → (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))))
352, 15, 34sylc 63 . . . 4 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → ((¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊)))
3635impr 647 . . 3 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ ((𝑃𝐴𝑄𝐴) ∧ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊))) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
371, 36sylan2b 491 . 2 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
38373impb 1252 1 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  LHypclh 34288  LDilcldil 34404  LTrncltrn 34405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-ltrn 34409 This theorem is referenced by:  ltrncnv  34450  trlval2  34468  cdlemg14f  34959  cdlemg14g  34960
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