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Theorem ltrnu 35725
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 882 . . 3 (((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ↔ ((𝑃𝐴𝑄𝐴) ∧ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊)))
2 simpr 476 . . . . 5 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → (𝑃𝐴𝑄𝐴))
3 simplr 807 . . . . . 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 2651 . . . . . . . . 9 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10 ltrnu.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10isltrn 35723 . . . . . . . 8 ((𝐾𝑉𝑊𝐻) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
1211ad2antrr 762 . . . . . . 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 243 . . . . . 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 4688 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 𝑊𝑃 𝑊))
1716notbid 307 . . . . . . . 8 (𝑝 = 𝑃 → (¬ 𝑝 𝑊 ↔ ¬ 𝑃 𝑊))
1817anbi1d 741 . . . . . . 7 (𝑝 = 𝑃 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊)))
19 id 22 . . . . . . . . . 10 (𝑝 = 𝑃𝑝 = 𝑃)
20 fveq2 6229 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
2119, 20oveq12d 6708 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 (𝐹𝑝)) = (𝑃 (𝐹𝑃)))
2221oveq1d 6705 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑃 (𝐹𝑃)) 𝑊))
2322eqeq1d 2653 . . . . . . 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 4688 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑞 𝑊𝑄 𝑊))
2625notbid 307 . . . . . . . 8 (𝑞 = 𝑄 → (¬ 𝑞 𝑊 ↔ ¬ 𝑄 𝑊))
2726anbi2d 740 . . . . . . 7 (𝑞 = 𝑄 → ((¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊)))
28 id 22 . . . . . . . . . 10 (𝑞 = 𝑄𝑞 = 𝑄)
29 fveq2 6229 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
3028, 29oveq12d 6708 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑞 (𝐹𝑞)) = (𝑄 (𝐹𝑄)))
3130oveq1d 6705 . . . . . . . 8 (𝑞 = 𝑄 → ((𝑞 (𝐹𝑞)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
3231eqeq2d 2661 . . . . . . 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 3353 . . . . 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 65 . . . 4 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → ((¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊)))
3635impr 648 . . 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 1279 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  LHypclh 35588  LDilcldil 35704  LTrncltrn 35705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-ltrn 35709
This theorem is referenced by:  ltrncnv  35750  trlval2  35768  cdlemg14f  36258  cdlemg14g  36259
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