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Theorem isltrn2N 40103
Description: The predicate "is a lattice translation". Version of isltrn 40102 that considers only different 𝑝 and 𝑞. TODO: Can this eliminate some separate proofs for the 𝑝 = 𝑞 case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
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
isltrn2N ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((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 isltrn2N
StepHypRef Expression
1 ltrnset.l . . 3 = (le‘𝐾)
2 ltrnset.j . . 3 = (join‘𝐾)
3 ltrnset.m . . 3 = (meet‘𝐾)
4 ltrnset.a . . 3 𝐴 = (Atoms‘𝐾)
5 ltrnset.h . . 3 𝐻 = (LHyp‘𝐾)
6 ltrnset.d . . 3 𝐷 = ((LDil‘𝐾)‘𝑊)
7 ltrnset.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7isltrn 40102 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
9 3simpa 1147 . . . . . 6 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊))
109imim1i 63 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
11 3anass 1094 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
12 3anrot 1099 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞))
13 df-ne 2939 . . . . . . . . . 10 (𝑝𝑞 ↔ ¬ 𝑝 = 𝑞)
1413anbi1i 624 . . . . . . . . 9 ((𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1511, 12, 143bitr3i 301 . . . . . . . 8 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1615imbi1i 349 . . . . . . 7 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
17 impexp 450 . . . . . . 7 (((¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
1816, 17bitri 275 . . . . . 6 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
19 id 22 . . . . . . . . . 10 (𝑝 = 𝑞𝑝 = 𝑞)
20 fveq2 6907 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
2119, 20oveq12d 7449 . . . . . . . . 9 (𝑝 = 𝑞 → (𝑝 (𝐹𝑝)) = (𝑞 (𝐹𝑞)))
2221oveq1d 7446 . . . . . . . 8 (𝑝 = 𝑞 → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
2322a1d 25 . . . . . . 7 (𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
24 pm2.61 192 . . . . . . 7 ((𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ((¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
2523, 24ax-mp 5 . . . . . 6 ((¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2618, 25sylbi 217 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2710, 26impbii 209 . . . 4 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
28272ralbii 3126 . . 3 (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2928anbi2i 623 . 2 ((𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
308, 29bitrdi 287 1 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059   class class class wbr 5148  cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  LHypclh 39967  LDilcldil 40083  LTrncltrn 40084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-ltrn 40088
This theorem is referenced by: (None)
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