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Theorem 2atjm 36023
 Description: The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
Hypotheses
Ref Expression
2atjm.b 𝐵 = (Base‘𝐾)
2atjm.l = (le‘𝐾)
2atjm.j = (join‘𝐾)
2atjm.m = (meet‘𝐾)
2atjm.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2atjm ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = 𝑃)

Proof of Theorem 2atjm
StepHypRef Expression
1 hllat 35941 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1113 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ Lat)
3 simp21 1186 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐴)
4 2atjm.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2atjm.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 35867 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝐵)
8 simp22 1187 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐴)
94, 5atbase 35867 . . . . . 6 (𝑄𝐴𝑄𝐵)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑄𝐵)
11 2atjm.l . . . . . 6 = (le‘𝐾)
12 2atjm.j . . . . . 6 = (join‘𝐾)
134, 11, 12latlej1 17528 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
142, 7, 10, 13syl3anc 1351 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 (𝑃 𝑄))
15 simp3l 1181 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 𝑋)
16 simp1 1116 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ HL)
174, 12, 5hlatjcl 35945 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
1816, 3, 8, 17syl3anc 1351 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
19 simp23 1188 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋𝐵)
20 2atjm.m . . . . . 6 = (meet‘𝐾)
214, 11, 20latlem12 17546 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑃 (𝑃 𝑄) ∧ 𝑃 𝑋) ↔ 𝑃 ((𝑃 𝑄) 𝑋)))
222, 7, 18, 19, 21syl13anc 1352 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 (𝑃 𝑄) ∧ 𝑃 𝑋) ↔ 𝑃 ((𝑃 𝑄) 𝑋)))
2314, 15, 22mpbi2and 699 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 ((𝑃 𝑄) 𝑋))
24 hlatl 35938 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25243ad2ant1 1113 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝐾 ∈ AtLat)
264, 20latmcom 17543 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑃 𝑄)))
272, 18, 19, 26syl3anc 1351 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = (𝑋 (𝑃 𝑄)))
2819, 3, 83jca 1108 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋𝐵𝑃𝐴𝑄𝐴))
29 nbrne2 4949 . . . . . . 7 ((𝑃 𝑋 ∧ ¬ 𝑄 𝑋) → 𝑃𝑄)
30293ad2ant3 1115 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
31 simp3r 1182 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ¬ 𝑄 𝑋)
324, 12latjcl 17519 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → (𝑋 𝑄) ∈ 𝐵)
332, 19, 10, 32syl3anc 1351 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋 𝑄) ∈ 𝐵)
344, 11, 12latlej1 17528 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑄𝐵) → 𝑋 (𝑋 𝑄))
352, 19, 10, 34syl3anc 1351 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑋 (𝑋 𝑄))
364, 11, 2, 7, 19, 33, 15, 35lattrd 17526 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 (𝑋 𝑄))
374, 11, 12, 20, 5cvrat3 36020 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴))
3837imp 398 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄))) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
3916, 28, 30, 31, 36, 38syl23anc 1357 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑋 (𝑃 𝑄)) ∈ 𝐴)
4027, 39eqeltrd 2867 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
4111, 5atcmp 35889 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑄) 𝑋) ∈ 𝐴) → (𝑃 ((𝑃 𝑄) 𝑋) ↔ 𝑃 = ((𝑃 𝑄) 𝑋)))
4225, 3, 40, 41syl3anc 1351 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑃 ((𝑃 𝑄) 𝑋) ↔ 𝑃 = ((𝑃 𝑄) 𝑋)))
4323, 42mpbid 224 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 = ((𝑃 𝑄) 𝑋))
4443eqcomd 2785 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ((𝑃 𝑄) 𝑋) = 𝑃)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2968   class class class wbr 4929  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  meetcmee 17413  Latclat 17513  Atomscatm 35841  AtLatcal 35842  HLchlt 35928 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-lat 17514  df-clat 17576  df-oposet 35754  df-ol 35756  df-oml 35757  df-covers 35844  df-ats 35845  df-atl 35876  df-cvlat 35900  df-hlat 35929 This theorem is referenced by:  atbtwn  36024  dalem24  36275  dalem25  36276
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