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Theorem cdlemg7fvbwN 38608
Description: Properties of a translation of an element not under 𝑊. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 38503? 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 2738 . . . 4 (join‘𝐾) = (join‘𝐾)
4 eqid 2738 . . . 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 38025 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
873adant3 1131 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → ∃𝑟𝐴𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋))
9 simp11 1202 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 simp2 1136 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑟𝐴)
11 simp3l 1200 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ 𝑟 𝑊)
1210, 11jca 512 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑟𝐴 ∧ ¬ 𝑟 𝑊))
13 simp12 1203 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
14 simp13 1204 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐹𝑇)
15 simp3r 1201 . . . . . 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 38600 . . . . . 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 1378 . . . . 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 1283 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ HL)
2019hllatd 37365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
212, 5, 6, 16ltrnel 38140 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊)) → ((𝐹𝑟) ∈ 𝐴 ∧ ¬ (𝐹𝑟) 𝑊))
2221simpld 495 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊)) → (𝐹𝑟) ∈ 𝐴)
239, 14, 12, 22syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑟) ∈ 𝐴)
241, 5atbase 37290 . . . . . . 7 ((𝐹𝑟) ∈ 𝐴 → (𝐹𝑟) ∈ 𝐵)
2523, 24syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑟) ∈ 𝐵)
26 simp12l 1285 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑋𝐵)
27 simp11r 1284 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊𝐻)
281, 6lhpbase 37999 . . . . . . . 8 (𝑊𝐻𝑊𝐵)
2927, 28syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → 𝑊𝐵)
301, 4latmcl 18147 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵)
3120, 26, 29, 30syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝑋(meet‘𝐾)𝑊) ∈ 𝐵)
321, 3latjcl 18146 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝑟) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) → ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) ∈ 𝐵)
3320, 25, 31, 32syl3anc 1370 . . . . 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)
3418, 33eqeltrd 2839 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → (𝐹𝑋) ∈ 𝐵)
3521simprd 496 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊)) → ¬ (𝐹𝑟) 𝑊)
369, 14, 12, 35syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ (𝐹𝑟) 𝑊)
371, 2, 3latlej1 18155 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑟) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)𝑊) ∈ 𝐵) → (𝐹𝑟) ((𝐹𝑟)(join‘𝐾)(𝑋(meet‘𝐾)𝑊)))
3820, 25, 31, 37syl3anc 1370 . . . . . . 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)))
391, 2lattr 18151 . . . . . . . 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))
4020, 25, 33, 29, 39syl13anc 1371 . . . . . . 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))
4138, 40mpand 692 . . . . . 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))
4236, 41mtod 197 . . . . 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)
4318breq1d 5085 . . . . 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))
4442, 43mtbird 325 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ¬ (𝐹𝑋) 𝑊)
4534, 44jca 512 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) ∧ 𝑟𝐴 ∧ (¬ 𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋)) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊))
4645rexlimdv3a 3214 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → (∃𝑟𝐴𝑟 𝑊 ∧ (𝑟(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊)))
478, 46mpd 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 396  w3a 1086   = wceq 1539  wcel 2106  wrex 3065   class class class wbr 5075  cfv 6428  (class class class)co 7269  Basecbs 16901  lecple 16958  joincjn 18018  meetcmee 18019  Latclat 18138  Atomscatm 37264  HLchlt 37351  LHypclh 37985  LTrncltrn 38102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580  ax-riotaBAD 36954
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-iin 4929  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-undef 8078  df-map 8606  df-proset 18002  df-poset 18020  df-plt 18037  df-lub 18053  df-glb 18054  df-join 18055  df-meet 18056  df-p0 18132  df-p1 18133  df-lat 18139  df-clat 18206  df-oposet 37177  df-ol 37179  df-oml 37180  df-covers 37267  df-ats 37268  df-atl 37299  df-cvlat 37323  df-hlat 37352  df-llines 37499  df-lplanes 37500  df-lvols 37501  df-lines 37502  df-psubsp 37504  df-pmap 37505  df-padd 37797  df-lhyp 37989  df-laut 37990  df-ldil 38105  df-ltrn 38106  df-trl 38160
This theorem is referenced by:  cdlemg7fvN  38625
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