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Theorem 2llnma1b 39743
Description: Generalization of 2llnma1 39744. (Contributed by NM, 26-Apr-2013.)
Hypotheses
Ref Expression
2llnma1b.b 𝐵 = (Base‘𝐾)
2llnma1b.l = (le‘𝐾)
2llnma1b.j = (join‘𝐾)
2llnma1b.m = (meet‘𝐾)
2llnma1b.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
2llnma1b ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)

Proof of Theorem 2llnma1b
StepHypRef Expression
1 hllat 39319 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1133 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp22 1207 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐴)
4 2llnma1b.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2llnma1b.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 39245 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐵)
8 simp21 1206 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑋𝐵)
9 2llnma1b.l . . . . . 6 = (le‘𝐾)
10 2llnma1b.j . . . . . 6 = (join‘𝐾)
114, 9, 10latlej1 18518 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → 𝑃 (𝑃 𝑋))
122, 7, 8, 11syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑋))
13 simp23 1208 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐴)
144, 5atbase 39245 . . . . . 6 (𝑄𝐴𝑄𝐵)
1513, 14syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐵)
164, 9, 10latlej1 18518 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
172, 7, 15, 16syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑄))
184, 10latjcl 18509 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
192, 7, 8, 18syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ∈ 𝐵)
20 simp1 1136 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ HL)
214, 10, 5hlatjcl 39323 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
2220, 3, 13, 21syl3anc 1371 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
23 2llnma1b.m . . . . . 6 = (meet‘𝐾)
244, 9, 23latlem12 18536 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑄) ∈ 𝐵)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
252, 7, 19, 22, 24syl13anc 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
2612, 17, 25mpbi2and 711 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) (𝑃 𝑄)))
27 hlatl 39316 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28273ad2ant1 1133 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ AtLat)
29 simp3 1138 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ¬ 𝑄 (𝑃 𝑋))
30 nbrne2 5186 . . . . . 6 ((𝑃 (𝑃 𝑋) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
3112, 29, 30syl2anc 583 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
324, 10latjcl 18509 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
332, 19, 15, 32syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
344, 9, 10latlej1 18518 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
352, 19, 15, 34syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
364, 9, 2, 7, 19, 33, 12, 35lattrd 18516 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) 𝑄))
374, 9, 10, 23, 5cvrat3 39399 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴))
38373impia 1117 . . . . 5 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄))) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
3920, 19, 3, 13, 31, 29, 36, 38syl133anc 1393 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
409, 5atcmp 39267 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4128, 3, 39, 40syl3anc 1371 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4226, 41mpbid 232 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 = ((𝑃 𝑋) (𝑃 𝑄)))
4342eqcomd 2746 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wne 2946   class class class wbr 5166  cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  meetcmee 18382  Latclat 18501  Atomscatm 39219  AtLatcal 39220  HLchlt 39306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307
This theorem is referenced by:  2llnma1  39744  cdlemg4  40574  cdlemkfid1N  40878
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