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Theorem 2llnma1b 36479
Description: Generalization of 2llnma1 36480. (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 36056 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1126 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp22 1200 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐴)
4 2llnma1b.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2llnma1b.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 35982 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐵)
8 simp21 1199 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑋𝐵)
9 2llnma1b.l . . . . . 6 = (le‘𝐾)
10 2llnma1b.j . . . . . 6 = (join‘𝐾)
114, 9, 10latlej1 17504 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → 𝑃 (𝑃 𝑋))
122, 7, 8, 11syl3anc 1364 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑋))
13 simp23 1201 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐴)
144, 5atbase 35982 . . . . . 6 (𝑄𝐴𝑄𝐵)
1513, 14syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐵)
164, 9, 10latlej1 17504 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
172, 7, 15, 16syl3anc 1364 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑄))
184, 10latjcl 17495 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
192, 7, 8, 18syl3anc 1364 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ∈ 𝐵)
20 simp1 1129 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ HL)
214, 10, 5hlatjcl 36060 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
2220, 3, 13, 21syl3anc 1364 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
23 2llnma1b.m . . . . . 6 = (meet‘𝐾)
244, 9, 23latlem12 17522 . . . . 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 1365 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
2612, 17, 25mpbi2and 708 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) (𝑃 𝑄)))
27 hlatl 36053 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28273ad2ant1 1126 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ AtLat)
29 simp3 1131 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ¬ 𝑄 (𝑃 𝑋))
30 nbrne2 4986 . . . . . 6 ((𝑃 (𝑃 𝑋) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
3112, 29, 30syl2anc 584 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
324, 10latjcl 17495 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
332, 19, 15, 32syl3anc 1364 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
344, 9, 10latlej1 17504 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
352, 19, 15, 34syl3anc 1364 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
364, 9, 2, 7, 19, 33, 12, 35lattrd 17502 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) 𝑄))
374, 9, 10, 23, 5cvrat3 36135 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴))
38373impia 1110 . . . . 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 1386 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
409, 5atcmp 36004 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4128, 3, 39, 40syl3anc 1364 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4226, 41mpbid 233 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 = ((𝑃 𝑋) (𝑃 𝑄)))
4342eqcomd 2801 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984   class class class wbr 4966  cfv 6230  (class class class)co 7021  Basecbs 16317  lecple 16406  joincjn 17388  meetcmee 17389  Latclat 17489  Atomscatm 35956  AtLatcal 35957  HLchlt 36043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-proset 17372  df-poset 17390  df-plt 17402  df-lub 17418  df-glb 17419  df-join 17420  df-meet 17421  df-p0 17483  df-lat 17490  df-clat 17552  df-oposet 35869  df-ol 35871  df-oml 35872  df-covers 35959  df-ats 35960  df-atl 35991  df-cvlat 36015  df-hlat 36044
This theorem is referenced by:  2llnma1  36480  cdlemg4  37310  cdlemkfid1N  37614
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