Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2atm Structured version   Visualization version   GIF version

Theorem 2atm 34328
Description: An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)
Hypotheses
Ref Expression
2atm.l = (le‘𝐾)
2atm.j = (join‘𝐾)
2atm.m = (meet‘𝐾)
2atm.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2atm (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 = ((𝑃 𝑄) (𝑅 𝑆)))

Proof of Theorem 2atm
StepHypRef Expression
1 simp31 1095 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑃 𝑄))
2 simp32 1096 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑅 𝑆))
3 simp11 1089 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ HL)
4 hllat 34165 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ Lat)
6 simp23 1094 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇𝐴)
7 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
8 2atm.a . . . . . 6 𝐴 = (Atoms‘𝐾)
97, 8atbase 34091 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
106, 9syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 ∈ (Base‘𝐾))
11 simp12 1090 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑃𝐴)
127, 8atbase 34091 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1311, 12syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑃 ∈ (Base‘𝐾))
14 simp13 1091 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑄𝐴)
157, 8atbase 34091 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1614, 15syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑄 ∈ (Base‘𝐾))
17 2atm.j . . . . . 6 = (join‘𝐾)
187, 17latjcl 16983 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
195, 13, 16, 18syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ∈ (Base‘𝐾))
20 simp21 1092 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑅𝐴)
21 simp22 1093 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑆𝐴)
227, 17, 8hlatjcl 34168 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
233, 20, 21, 22syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑅 𝑆) ∈ (Base‘𝐾))
24 2atm.l . . . . 5 = (le‘𝐾)
25 2atm.m . . . . 5 = (meet‘𝐾)
267, 24, 25latlem12 17010 . . . 4 ((𝐾 ∈ Lat ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾))) → ((𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆)) ↔ 𝑇 ((𝑃 𝑄) (𝑅 𝑆))))
275, 10, 19, 23, 26syl13anc 1325 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆)) ↔ 𝑇 ((𝑃 𝑄) (𝑅 𝑆))))
281, 2, 27mpbi2and 955 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 ((𝑃 𝑄) (𝑅 𝑆)))
29 hlatl 34162 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
303, 29syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ AtLat)
317, 25latmcl 16984 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
325, 19, 23, 31syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
33 eqid 2621 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
347, 24, 33, 8atlen0 34112 . . . . . 6 (((𝐾 ∈ AtLat ∧ ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾) ∧ 𝑇𝐴) ∧ 𝑇 ((𝑃 𝑄) (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ≠ (0.‘𝐾))
3530, 32, 6, 28, 34syl31anc 1326 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ≠ (0.‘𝐾))
3635neneqd 2795 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ¬ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾))
37 simp33 1097 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ≠ (𝑅 𝑆))
3817, 25, 33, 82atmat0 34327 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴 ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ∨ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾)))
393, 11, 14, 20, 21, 37, 38syl33anc 1338 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ∨ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾)))
4039ord 392 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (¬ ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 → ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾)))
4136, 40mt3d 140 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴)
4224, 8atcmp 34113 . . 3 ((𝐾 ∈ AtLat ∧ 𝑇𝐴 ∧ ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴) → (𝑇 ((𝑃 𝑄) (𝑅 𝑆)) ↔ 𝑇 = ((𝑃 𝑄) (𝑅 𝑆))))
4330, 6, 41, 42syl3anc 1323 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑇 ((𝑃 𝑄) (𝑅 𝑆)) ↔ 𝑇 = ((𝑃 𝑄) (𝑅 𝑆))))
4428, 43mpbid 222 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 = ((𝑃 𝑄) (𝑅 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  meetcmee 16877  0.cp0 16969  Latclat 16977  Atomscatm 34065  AtLatcal 34066  HLchlt 34152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-clat 17040  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-llines 34299
This theorem is referenced by:  cdlemk12  35653  cdlemk12u  35675  cdlemk47  35752
  Copyright terms: Public domain W3C validator