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Theorem isltrn2N 37361
Description: The predicate "is a lattice translation". Version of isltrn 37360 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 37360 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
9 3simpa 1145 . . . . . 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 1092 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
12 3anrot 1097 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞))
13 df-ne 3015 . . . . . . . . . 10 (𝑝𝑞 ↔ ¬ 𝑝 = 𝑞)
1413anbi1i 626 . . . . . . . . 9 ((𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1511, 12, 143bitr3i 304 . . . . . . . 8 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1615imbi1i 353 . . . . . . 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 454 . . . . . . 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 278 . . . . . 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 6661 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
2119, 20oveq12d 7167 . . . . . . . . 9 (𝑝 = 𝑞 → (𝑝 (𝐹𝑝)) = (𝑞 (𝐹𝑞)))
2221oveq1d 7164 . . . . . . . 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 pm2.61 195 . . . . . . 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 220 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2710, 26impbii 212 . . . 4 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
28272ralbii 3161 . . 3 (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2928anbi2i 625 . 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, 29syl6bb 290 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133   class class class wbr 5052  cfv 6343  (class class class)co 7149  lecple 16572  joincjn 17554  meetcmee 17555  Atomscatm 36504  LHypclh 37225  LDilcldil 37341  LTrncltrn 37342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-ltrn 37346
This theorem is referenced by: (None)
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