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Theorem cdlemg7fvbwN 36409
Description: Properties of a translation of an element not under 𝑊. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 36304? Done with a *ltrn* theorem? (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg4.l = (le‘𝐾)
cdlemg4.a 𝐴 = (Atoms‘𝐾)
cdlemg4.h 𝐻 = (LHyp‘𝐾)
cdlemg4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
cdlemg7fvbwN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊))

Proof of Theorem cdlemg7fvbwN
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 cdlemg4.b . . . 4 𝐵 = (Base‘𝐾)
2 cdlemg4.l . . . 4 = (le‘𝐾)
3 eqid 2770 . . . 4 (join‘𝐾) = (join‘𝐾)
4 eqid 2770 . . . 4 (meet‘𝐾) = (meet‘𝐾)
5 cdlemg4.a . . . 4 𝐴 = (Atoms‘𝐾)
6 cdlemg4.h . . . 4 𝐻 = (LHyp‘𝐾)
71, 2, 3, 4, 5, 6lhpmcvr2 35825 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
873adant3 1125 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
9 simp11 1244 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 simp2 1130 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑟𝐴)
11 simp3l 1242 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ 𝑟 𝑊)
1210, 11jca 495 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟𝐴 ∧ ¬ 𝑟 𝑊))
13 simp12 1245 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
14 simp13 1246 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐹𝑇)
15 simp3r 1243 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)
16 cdlemg4.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
176, 16, 2, 3, 5, 4, 1cdlemg2fv 36401 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
189, 12, 13, 14, 15, 17syl122anc 1484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
19 simp11l 1367 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ HL)
20 hllat 35165 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2119, 20syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
222, 5, 6, 16ltrnel 35940 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊)) → ((𝐹𝑟) ∈ 𝐴 ∧ ¬ (𝐹𝑟) 𝑊))
2322simpld 476 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊)) → (𝐹𝑟) ∈ 𝐴)
249, 14, 12, 23syl3anc 1475 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑟) ∈ 𝐴)
251, 5atbase 35091 . . . . . . 7 ((𝐹𝑟) ∈ 𝐴 → (𝐹𝑟) ∈ 𝐵)
2624, 25syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑟) ∈ 𝐵)
27 simp12l 1369 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑋𝐵)
28 simp11r 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊𝐻)
291, 6lhpbase 35799 . . . . . . . 8 (𝑊𝐻𝑊𝐵)
3028, 29syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊𝐵)
311, 4latmcl 17259 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵)
3221, 27, 30, 31syl3anc 1475 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵)
331, 3latjcl 17258 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝑟) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) → ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵)
3421, 26, 32, 33syl3anc 1475 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵)
3518, 34eqeltrd 2849 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑋) ∈ 𝐵)
3622simprd 477 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊)) → ¬ (𝐹𝑟) 𝑊)
379, 14, 12, 36syl3anc 1475 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ (𝐹𝑟) 𝑊)
381, 2, 3latlej1 17267 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑟) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) → (𝐹𝑟) ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
3921, 26, 32, 38syl3anc 1475 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑟) ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
401, 2lattr 17263 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝐹𝑟) ∈ 𝐵 ∧ ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵𝑊𝐵)) → (((𝐹𝑟) ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∧ ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) 𝑊) → (𝐹𝑟) 𝑊))
4121, 26, 34, 30, 40syl13anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((𝐹𝑟) ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∧ ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) 𝑊) → (𝐹𝑟) 𝑊))
4239, 41mpand 667 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) 𝑊 → (𝐹𝑟) 𝑊))
4337, 42mtod 189 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) 𝑊)
4418breq1d 4794 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹𝑋) 𝑊 ↔ ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) 𝑊))
4543, 44mtbird 314 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ (𝐹𝑋) 𝑊)
4635, 45jca 495 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊))
4746rexlimdv3a 3180 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊)))
488, 47mpd 15 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wrex 3061   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  joincjn 17151  meetcmee 17152  Latclat 17252  Atomscatm 35065  HLchlt 35152  LHypclh 35785  LTrncltrn 35902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-undef 7550  df-map 8010  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789  df-laut 35790  df-ldil 35905  df-ltrn 35906  df-trl 35961
This theorem is referenced by:  cdlemg7fvN  36426
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