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Theorem 2llnma1b 35567
Description: Generalization of 2llnma1 35568. (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 35145 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1127 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp22 1247 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐴)
4 2llnma1b.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2llnma1b.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 35071 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐵)
8 simp21 1246 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑋𝐵)
9 2llnma1b.l . . . . . 6 = (le‘𝐾)
10 2llnma1b.j . . . . . 6 = (join‘𝐾)
114, 9, 10latlej1 17253 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → 𝑃 (𝑃 𝑋))
122, 7, 8, 11syl3anc 1473 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑋))
13 simp23 1248 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐴)
144, 5atbase 35071 . . . . . 6 (𝑄𝐴𝑄𝐵)
1513, 14syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐵)
164, 9, 10latlej1 17253 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
172, 7, 15, 16syl3anc 1473 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑄))
184, 10latjcl 17244 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
192, 7, 8, 18syl3anc 1473 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ∈ 𝐵)
20 simp1 1130 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ HL)
214, 10, 5hlatjcl 35148 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
2220, 3, 13, 21syl3anc 1473 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
23 2llnma1b.m . . . . . 6 = (meet‘𝐾)
244, 9, 23latlem12 17271 . . . . 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 1475 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
2612, 17, 25mpbi2and 994 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) (𝑃 𝑄)))
27 hlatl 35142 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28273ad2ant1 1127 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ AtLat)
29 simp3 1132 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ¬ 𝑄 (𝑃 𝑋))
30 nbrne2 4816 . . . . . 6 ((𝑃 (𝑃 𝑋) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
3112, 29, 30syl2anc 696 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
324, 10latjcl 17244 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
332, 19, 15, 32syl3anc 1473 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
344, 9, 10latlej1 17253 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
352, 19, 15, 34syl3anc 1473 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
364, 9, 2, 7, 19, 33, 12, 35lattrd 17251 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) 𝑄))
374, 9, 10, 23, 5cvrat3 35223 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴))
38373impia 1109 . . . . 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 1496 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
409, 5atcmp 35093 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4128, 3, 39, 40syl3anc 1473 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4226, 41mpbid 222 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 = ((𝑃 𝑋) (𝑃 𝑄)))
4342eqcomd 2758 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924   class class class wbr 4796  cfv 6041  (class class class)co 6805  Basecbs 16051  lecple 16142  joincjn 17137  meetcmee 17138  Latclat 17238  Atomscatm 35045  AtLatcal 35046  HLchlt 35132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-preset 17121  df-poset 17139  df-plt 17151  df-lub 17167  df-glb 17168  df-join 17169  df-meet 17170  df-p0 17232  df-lat 17239  df-clat 17301  df-oposet 34958  df-ol 34960  df-oml 34961  df-covers 35048  df-ats 35049  df-atl 35080  df-cvlat 35104  df-hlat 35133
This theorem is referenced by:  2llnma1  35568  cdlemg4  36399  cdlemkfid1N  36703
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