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Theorem 2llnma1b 38645
Description: Generalization of 2llnma1 38646. (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 38221 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
213ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝐾 ∈ Lat)
3 simp22 1207 . . . . . 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 38147 . . . . . 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 1206 . . . . 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 18397 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ 𝑃 ≀ (𝑃 ∨ 𝑋))
122, 7, 8, 11syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑋))
13 simp23 1208 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑄 ∈ 𝐴)
144, 5atbase 38147 . . . . . 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 18397 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
172, 7, 15, 16syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
184, 10latjcl 18388 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∨ 𝑋) ∈ 𝐡)
192, 7, 8, 18syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ (𝑃 ∨ 𝑋) ∈ 𝐡)
20 simp1 1136 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝐾 ∈ HL)
214, 10, 5hlatjcl 38225 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2220, 3, 13, 21syl3anc 1371 . . . . 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 18415 . . . . 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 1372 . . . 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 38218 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
28273ad2ant1 1133 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝐾 ∈ AtLat)
29 simp3 1138 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋))
30 nbrne2 5167 . . . . . 6 ((𝑃 ≀ (𝑃 ∨ 𝑋) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 β‰  𝑄)
3112, 29, 30syl2anc 584 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 β‰  𝑄)
324, 10latjcl 18388 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐡)
332, 19, 15, 32syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐡)
344, 9, 10latlej1 18397 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑋) ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))
352, 19, 15, 34syl3anc 1371 . . . . . 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 18395 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ 𝑃 ≀ ((𝑃 ∨ 𝑋) ∨ 𝑄))
374, 9, 10, 23, 5cvrat3 38301 . . . . . 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 1117 . . . . 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 1393 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ 𝑋)) β†’ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
409, 5atcmp 38169 . . . 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 1371 . . 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 2738 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  Latclat 18380  Atomscatm 38121  AtLatcal 38122  HLchlt 38208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  2llnma1  38646  cdlemg4  39476  cdlemkfid1N  39780
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