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Theorem 2atm 39510
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 1208 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑃 𝑄))
2 simp32 1209 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑅 𝑆))
3 simp11 1202 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ HL)
43hllatd 39346 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ Lat)
5 simp23 1207 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇𝐴)
6 eqid 2735 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 2atm.a . . . . . 6 𝐴 = (Atoms‘𝐾)
86, 7atbase 39271 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
95, 8syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 ∈ (Base‘𝐾))
10 simp12 1203 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑃𝐴)
116, 7atbase 39271 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1210, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑃 ∈ (Base‘𝐾))
13 simp13 1204 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑄𝐴)
146, 7atbase 39271 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1513, 14syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑄 ∈ (Base‘𝐾))
16 2atm.j . . . . . 6 = (join‘𝐾)
176, 16latjcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
184, 12, 15, 17syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ∈ (Base‘𝐾))
19 simp21 1205 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑅𝐴)
20 simp22 1206 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑆𝐴)
216, 16, 7hlatjcl 39349 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
223, 19, 20, 21syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑅 𝑆) ∈ (Base‘𝐾))
23 2atm.l . . . . 5 = (le‘𝐾)
24 2atm.m . . . . 5 = (meet‘𝐾)
256, 23, 24latlem12 18524 . . . 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))))
264, 9, 18, 22, 25syl13anc 1371 . . 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))))
271, 2, 26mpbi2and 712 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 ((𝑃 𝑄) (𝑅 𝑆)))
28 hlatl 39342 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
293, 28syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ AtLat)
306, 24latmcl 18498 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
314, 18, 22, 30syl3anc 1370 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
32 eqid 2735 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
336, 23, 32, 7atlen0 39292 . . . . . 6 (((𝐾 ∈ AtLat ∧ ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾) ∧ 𝑇𝐴) ∧ 𝑇 ((𝑃 𝑄) (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ≠ (0.‘𝐾))
3429, 31, 5, 27, 33syl31anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ≠ (0.‘𝐾))
3534neneqd 2943 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ¬ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾))
36 simp33 1210 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ≠ (𝑅 𝑆))
3716, 24, 32, 72atmat0 39509 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴 ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴 ∨ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾)))
383, 10, 13, 19, 20, 36, 37syl33anc 1384 . . . . 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)))
3938ord 864 . . . 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)))
4035, 39mt3d 148 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴)
4123, 7atcmp 39293 . . 3 ((𝐾 ∈ AtLat ∧ 𝑇𝐴 ∧ ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴) → (𝑇 ((𝑃 𝑄) (𝑅 𝑆)) ↔ 𝑇 = ((𝑃 𝑄) (𝑅 𝑆))))
4229, 5, 40, 41syl3anc 1370 . 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))))
4327, 42mpbid 232 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 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  0.cp0 18481  Latclat 18489  Atomscatm 39245  AtLatcal 39246  HLchlt 39332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481
This theorem is referenced by:  cdlemk12  40833  cdlemk12u  40855  cdlemk47  40932
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