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Theorem isltrn2N 37136
Description: The predicate "is a lattice translation". Version of isltrn 37135 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 37135 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
9 3simpa 1140 . . . . . 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 1087 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
12 3anrot 1092 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞))
13 df-ne 3014 . . . . . . . . . 10 (𝑝𝑞 ↔ ¬ 𝑝 = 𝑞)
1413anbi1i 623 . . . . . . . . 9 ((𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1511, 12, 143bitr3i 302 . . . . . . . 8 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1615imbi1i 351 . . . . . . 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 451 . . . . . . 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 276 . . . . . 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 6663 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
2119, 20oveq12d 7163 . . . . . . . . 9 (𝑝 = 𝑞 → (𝑝 (𝐹𝑝)) = (𝑞 (𝐹𝑞)))
2221oveq1d 7160 . . . . . . . 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 pm2.61 193 . . . . . . 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 218 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2710, 26impbii 210 . . . 4 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
28272ralbii 3163 . . 3 (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2928anbi2i 622 . 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 288 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135   class class class wbr 5057  cfv 6348  (class class class)co 7145  lecple 16560  joincjn 17542  meetcmee 17543  Atomscatm 36279  LHypclh 37000  LDilcldil 37116  LTrncltrn 37117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-ltrn 37121
This theorem is referenced by: (None)
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