Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isltrn2N Structured version   Visualization version   GIF version

Theorem isltrn2N 35909
Description: The predicate "is a lattice translation". Version of isltrn 35908 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 35908 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
9 3simpa 1143 . . . . . 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 1081 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
12 3anrot 1087 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞))
13 df-ne 2933 . . . . . . . . . 10 (𝑝𝑞 ↔ ¬ 𝑝 = 𝑞)
1413anbi1i 733 . . . . . . . . 9 ((𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1511, 12, 143bitr3i 290 . . . . . . . 8 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1615imbi1i 338 . . . . . . 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 461 . . . . . . 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 264 . . . . . 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 6352 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
2119, 20oveq12d 6831 . . . . . . . . 9 (𝑝 = 𝑞 → (𝑝 (𝐹𝑝)) = (𝑞 (𝐹𝑞)))
2221oveq1d 6828 . . . . . . . 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 pm2.61 183 . . . . . . 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 207 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2710, 26impbii 199 . . . 4 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
28272ralbii 3119 . . 3 (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2928anbi2i 732 . 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 276 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  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: (None)
  Copyright terms: Public domain W3C validator