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Theorem 2atm 39529
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 1210 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑃 𝑄))
2 simp32 1211 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇 (𝑅 𝑆))
3 simp11 1204 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ HL)
43hllatd 39365 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝐾 ∈ Lat)
5 simp23 1209 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑇𝐴)
6 eqid 2737 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
7 2atm.a . . . . . 6 𝐴 = (Atoms‘𝐾)
86, 7atbase 39290 . . . . 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 1205 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑃𝐴)
116, 7atbase 39290 . . . . . 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 1206 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑄𝐴)
146, 7atbase 39290 . . . . . 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 18484 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
184, 12, 15, 17syl3anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ∈ (Base‘𝐾))
19 simp21 1207 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑅𝐴)
20 simp22 1208 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → 𝑆𝐴)
216, 16, 7hlatjcl 39368 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
223, 19, 20, 21syl3anc 1373 . . . 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 18511 . . . 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 1374 . . 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 39361 . . . 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 18485 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
314, 18, 22, 30syl3anc 1373 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ∈ (Base‘𝐾))
32 eqid 2737 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
336, 23, 32, 7atlen0 39311 . . . . . 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 1375 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ((𝑃 𝑄) (𝑅 𝑆)) ≠ (0.‘𝐾))
3534neneqd 2945 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → ¬ ((𝑃 𝑄) (𝑅 𝑆)) = (0.‘𝐾))
36 simp33 1212 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑇 (𝑃 𝑄) ∧ 𝑇 (𝑅 𝑆) ∧ (𝑃 𝑄) ≠ (𝑅 𝑆))) → (𝑃 𝑄) ≠ (𝑅 𝑆))
3716, 24, 32, 72atmat0 39528 . . . . . 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 1387 . . . . 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 865 . . . 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 39312 . . 3 ((𝐾 ∈ AtLat ∧ 𝑇𝐴 ∧ ((𝑃 𝑄) (𝑅 𝑆)) ∈ 𝐴) → (𝑇 ((𝑃 𝑄) (𝑅 𝑆)) ↔ 𝑇 = ((𝑃 𝑄) (𝑅 𝑆))))
4229, 5, 40, 41syl3anc 1373 . 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  Latclat 18476  Atomscatm 39264  AtLatcal 39265  HLchlt 39351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500
This theorem is referenced by:  cdlemk12  40852  cdlemk12u  40874  cdlemk47  40951
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