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Theorem 2llnma1b 39299
Description: Generalization of 2llnma1 39300. (Contributed by NM, 26-Apr-2013.)
Hypotheses
Ref Expression
2llnma1b.b 𝐡 = (Baseβ€˜πΎ)
2llnma1b.l ≀ = (leβ€˜πΎ)
2llnma1b.j ∨ = (joinβ€˜πΎ)
2llnma1b.m ∧ = (meetβ€˜πΎ)
2llnma1b.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
2llnma1b ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) = 𝑃)
Proof of Theorem 2llnma1b
StepHypRef Expression
1 hllat 38875 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1130 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝐾 ∈ Lat)
3 simp22 1204 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ∈ 𝐴)
4 2llnma1b.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
5 2llnma1b.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38801 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ∈ 𝐡)
8 simp21 1203 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑋 ∈ 𝐡)
9 2llnma1b.l . . . . . 6 ≀ = (leβ€˜πΎ)
10 2llnma1b.j . . . . . 6 ∨ = (joinβ€˜πΎ)
114, 9, 10latlej1 18449 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ 𝑃 ≀ (𝑃 ∨ 𝑋))
122, 7, 8, 11syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑋))
13 simp23 1205 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑄 ∈ 𝐴)
144, 5atbase 38801 . . . . . 6 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1513, 14syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑄 ∈ 𝐡)
164, 9, 10latlej1 18449 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
172, 7, 15, 16syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
184, 10latjcl 18440 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∨ 𝑋) ∈ 𝐡)
192, 7, 8, 18syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ (𝑃 ∨ 𝑋) ∈ 𝐡)
20 simp1 1133 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝐾 ∈ HL)
214, 10, 5hlatjcl 38879 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2220, 3, 13, 21syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
23 2llnma1b.m . . . . . 6 ∧ = (meetβ€˜πΎ)
244, 9, 23latlem12 18467 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ (𝑃 ∨ 𝑋) ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑋) ∧ 𝑃 ≀ (𝑃 ∨ 𝑄)) ↔ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
252, 7, 19, 22, 24syl13anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑋) ∧ 𝑃 ≀ (𝑃 ∨ 𝑄)) ↔ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
2612, 17, 25mpbi2and 710 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
27 hlatl 38872 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
28273ad2ant1 1130 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝐾 ∈ AtLat)
29 simp3 1135 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋))
30 nbrne2 5172 . . . . . 6 ((𝑃 ≀ (𝑃 ∨ 𝑋) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 β‰  𝑄)
3112, 29, 30syl2anc 582 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 β‰  𝑄)
324, 10latjcl 18440 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐡)
332, 19, 15, 32syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐡)
344, 9, 10latlej1 18449 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑋) ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))
352, 19, 15, 34syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ (𝑃 ∨ 𝑋) ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))
364, 9, 2, 7, 19, 33, 12, 35lattrd 18447 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))
374, 9, 10, 23, 5cvrat3 38955 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑋) ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋) ∧ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄)) β†’ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
38373impia 1114 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑋) ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋) ∧ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))) β†’ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
3920, 19, 3, 13, 31, 29, 36, 38syl133anc 1390 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
409, 5atcmp 38823 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) β†’ (𝑃 ≀ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ↔ 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
4128, 3, 39, 40syl3anc 1368 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ (𝑃 ≀ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ↔ 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
4226, 41mpbid 231 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
4342eqcomd 2734 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  lecple 17249  joincjn 18312  meetcmee 18313  Latclat 18432  Atomscatm 38775  AtLatcal 38776  HLchlt 38862
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18296  df-poset 18314  df-plt 18331  df-lub 18347  df-glb 18348  df-join 18349  df-meet 18350  df-p0 18426  df-lat 18433  df-clat 18500  df-oposet 38688  df-ol 38690  df-oml 38691  df-covers 38778  df-ats 38779  df-atl 38810  df-cvlat 38834  df-hlat 38863
This theorem is referenced by:  2llnma1  39300  cdlemg4  40130  cdlemkfid1N  40434
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