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Theorem 2atm 35336
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 1251 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑃 𝑄))
2 simp32 1252 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑅 𝑆))
3 simp11 1245 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ HL)
43hllatd 35173 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ Lat)
5 simp23 1250 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇𝐴)
6 eqid 2771 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 2atm.a . . . . . 6 𝐴 = (Atoms‘𝐾)
86, 7atbase 35098 . . . . 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 1246 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑃𝐴)
116, 7atbase 35098 . . . . . 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 1247 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑄𝐴)
146, 7atbase 35098 . . . . . 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 17259 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
184, 12, 15, 17syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ∈ (Base‘𝐾))
19 simp21 1248 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑅𝐴)
20 simp22 1249 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑆𝐴)
216, 16, 7hlatjcl 35176 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
223, 19, 20, 21syl3anc 1476 . . . 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 17286 . . . 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 1478 . . 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 691 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 ((𝑃 𝑄) (𝑅 𝑆)))
28 hlatl 35169 . . . 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 17260 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
314, 18, 22, 30syl3anc 1476 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
32 eqid 2771 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
336, 23, 32, 7atlen0 35119 . . . . . 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 1479 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ≠ (0.‘𝐾))
3534neneqd 2948 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ¬ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾))
36 simp33 1253 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ≠ (𝑅 𝑆))
3716, 24, 32, 72atmat0 35335 . . . . . 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 1491 . . . . 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 853 . . . 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 142 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴)
4123, 7atcmp 35120 . . 3 ((𝐾 ∈ AtLat ∧ 𝑇𝐴 ∧ ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴) → (𝑇 ((𝑃 𝑄) (𝑅 𝑆)) ↔ 𝑇 = ((𝑃 𝑄) (𝑅 𝑆))))
4229, 5, 40, 41syl3anc 1476 . 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 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  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943   class class class wbr 4787  cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  0.cp0 17245  Latclat 17253  Atomscatm 35072  AtLatcal 35073  HLchlt 35159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35307
This theorem is referenced by:  cdlemk12  36660  cdlemk12u  36682  cdlemk47  36759
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