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Theorem ltrnu 35910
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 900 . . 3 (((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ↔ ((𝑃𝐴𝑄𝐴) ∧ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊)))
2 simpr 479 . . . . 5 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → (𝑃𝐴𝑄𝐴))
3 simplr 809 . . . . . 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 2760 . . . . . . . . 9 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10 ltrnu.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
114, 5, 6, 7, 8, 9, 10isltrn 35908 . . . . . . . 8 ((𝐾𝑉𝑊𝐻) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
1211ad2antrr 764 . . . . . . 7 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑃𝐴𝑄𝐴)) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
13 simpr 479 . . . . . . 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 4807 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 𝑊𝑃 𝑊))
1716notbid 307 . . . . . . . 8 (𝑝 = 𝑃 → (¬ 𝑝 𝑊 ↔ ¬ 𝑃 𝑊))
1817anbi1d 743 . . . . . . 7 (𝑝 = 𝑃 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊)))
19 id 22 . . . . . . . . . 10 (𝑝 = 𝑃𝑝 = 𝑃)
20 fveq2 6352 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
2119, 20oveq12d 6831 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 (𝐹𝑝)) = (𝑃 (𝐹𝑃)))
2221oveq1d 6828 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑃 (𝐹𝑃)) 𝑊))
2322eqeq1d 2762 . . . . . . 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 4807 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑞 𝑊𝑄 𝑊))
2625notbid 307 . . . . . . . 8 (𝑞 = 𝑄 → (¬ 𝑞 𝑊 ↔ ¬ 𝑄 𝑊))
2726anbi2d 742 . . . . . . 7 (𝑞 = 𝑄 → ((¬ 𝑃 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊)))
28 id 22 . . . . . . . . . 10 (𝑞 = 𝑄𝑞 = 𝑄)
29 fveq2 6352 . . . . . . . . . 10 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
3028, 29oveq12d 6831 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑞 (𝐹𝑞)) = (𝑄 (𝐹𝑄)))
3130oveq1d 6828 . . . . . . . 8 (𝑞 = 𝑄 → ((𝑞 (𝐹𝑞)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
3231eqeq2d 2770 . . . . . . 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 3461 . . . . 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 650 . . 3 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ ((𝑃𝐴𝑄𝐴) ∧ (¬ 𝑃 𝑊 ∧ ¬ 𝑄 𝑊))) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
371, 36sylan2b 493 . 2 ((((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 (𝐹𝑃)) 𝑊) = ((𝑄 (𝐹𝑄)) 𝑊))
38373impb 1108 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 1072   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804  cfv 6049  (class class class)co 6813  lecple 16150  joincjn 17145  meetcmee 17146  Atomscatm 35053  LHypclh 35773  LDilcldil 35889  LTrncltrn 35890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-ltrn 35894
This theorem is referenced by:  ltrncnv  35935  trlval2  35953  cdlemg14f  36443  cdlemg14g  36444
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