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Theorem 2atjm 38820
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 𝐡 = (Baseβ€˜πΎ)
2atjm.l ≀ = (leβ€˜πΎ)
2atjm.j ∨ = (joinβ€˜πΎ)
2atjm.m ∧ = (meetβ€˜πΎ)
2atjm.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
2atjm ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)
Proof of Theorem 2atjm
StepHypRef Expression
1 hllat 38737 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1130 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
3 simp21 1203 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐴)
4 2atjm.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
5 2atjm.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38663 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐡)
8 simp22 1204 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
94, 5atbase 38663 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
108, 9syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐡)
11 2atjm.l . . . . . 6 ≀ = (leβ€˜πΎ)
12 2atjm.j . . . . . 6 ∨ = (joinβ€˜πΎ)
134, 11, 12latlej1 18409 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
142, 7, 10, 13syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
15 simp3l 1198 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ 𝑋)
16 simp1 1133 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
174, 12, 5hlatjcl 38741 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
1816, 3, 8, 17syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
19 simp23 1205 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
20 2atjm.m . . . . . 6 ∧ = (meetβ€˜πΎ)
214, 11, 20latlem12 18427 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 ≀ 𝑋) ↔ 𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
222, 7, 18, 19, 21syl13anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 ≀ 𝑋) ↔ 𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
2314, 15, 22mpbi2and 709 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋))
24 hlatl 38734 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
25243ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ AtLat)
264, 20latmcom 18424 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
272, 18, 19, 26syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
2819, 3, 83jca 1125 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
29 nbrne2 5159 . . . . . . 7 ((𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑃 β‰  𝑄)
30293ad2ant3 1132 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 β‰  𝑄)
31 simp3r 1199 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ Β¬ 𝑄 ≀ 𝑋)
324, 12latjcl 18400 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
332, 19, 10, 32syl3anc 1368 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
344, 11, 12latlej1 18409 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ 𝑋 ≀ (𝑋 ∨ 𝑄))
352, 19, 10, 34syl3anc 1368 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑋 ≀ (𝑋 ∨ 𝑄))
364, 11, 2, 7, 19, 33, 15, 35lattrd 18407 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ (𝑋 ∨ 𝑄))
374, 11, 12, 20, 5cvrat3 38817 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
3837imp 406 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
3916, 28, 30, 31, 36, 38syl23anc 1374 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
4027, 39eqeltrd 2825 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
4111, 5atcmp 38685 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) β†’ (𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)))
4225, 3, 40, 41syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑃 ≀ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)))
4323, 42mpbid 231 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋))
4443eqcomd 2730 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  lecple 17209  joincjn 18272  meetcmee 18273  Latclat 18392  Atomscatm 38637  AtLatcal 38638  HLchlt 38724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-lat 18393  df-clat 18460  df-oposet 38550  df-ol 38552  df-oml 38553  df-covers 38640  df-ats 38641  df-atl 38672  df-cvlat 38696  df-hlat 38725
This theorem is referenced by:  atbtwn  38821  dalem24  39072  dalem25  39073
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