Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnma1b Structured version   Visualization version   GIF version
Theorem 2llnma1b 39170
Description: Generalization of 2llnma1 39171. (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 38746 . . . . . 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 38672 . . . . . 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 18413 . . . . 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 38672 . . . . . 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 18413 . . . . 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 18404 . . . . . 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 38750 . . . . . 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 18431 . . . . 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 709 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
27 hlatl 38743 . . . . 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 5161 . . . . . 6 ((𝑃 ≀ (𝑃 ∨ 𝑋) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 β‰  𝑄)
3112, 29, 30syl2anc 583 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 β‰  𝑄)
324, 10latjcl 18404 . . . . . . 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 18413 . . . . . . 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 18411 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))
374, 9, 10, 23, 5cvrat3 38826 . . . . . 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 38694 . . . 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 2732 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  Latclat 18396  Atomscatm 38646  AtLatcal 38647  HLchlt 38733
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  2llnma1  39171  cdlemg4  40001  cdlemkfid1N  40305
  Copyright terms: Public domain W3C validator