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Theorem 2llnma1b 34538
Description: Generalization of 2llnma1 34539. (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 34116 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
213ad2ant1 1080 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ Lat)
3 simp22 1093 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐴)
4 2llnma1b.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 2llnma1b.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 34042 . . . . . 6 (𝑃𝐴𝑃𝐵)
73, 6syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝐵)
8 simp21 1092 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑋𝐵)
9 2llnma1b.l . . . . . 6 = (le‘𝐾)
10 2llnma1b.j . . . . . 6 = (join‘𝐾)
114, 9, 10latlej1 16976 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → 𝑃 (𝑃 𝑋))
122, 7, 8, 11syl3anc 1323 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑋))
13 simp23 1094 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐴)
144, 5atbase 34042 . . . . . 6 (𝑄𝐴𝑄𝐵)
1513, 14syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑄𝐵)
164, 9, 10latlej1 16976 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → 𝑃 (𝑃 𝑄))
172, 7, 15, 16syl3anc 1323 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 (𝑃 𝑄))
184, 10latjcl 16967 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
192, 7, 8, 18syl3anc 1323 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ∈ 𝐵)
20 simp1 1059 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ HL)
214, 10, 5hlatjcl 34119 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
2220, 3, 13, 21syl3anc 1323 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑄) ∈ 𝐵)
23 2llnma1b.m . . . . . 6 = (meet‘𝐾)
244, 9, 23latlem12 16994 . . . . 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 1325 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 (𝑃 𝑋) ∧ 𝑃 (𝑃 𝑄)) ↔ 𝑃 ((𝑃 𝑋) (𝑃 𝑄))))
2612, 17, 25mpbi2and 955 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) (𝑃 𝑄)))
27 hlatl 34113 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28273ad2ant1 1080 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝐾 ∈ AtLat)
29 simp3 1061 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ¬ 𝑄 (𝑃 𝑋))
30 nbrne2 4638 . . . . . 6 ((𝑃 (𝑃 𝑋) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
3112, 29, 30syl2anc 692 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃𝑄)
324, 10latjcl 16967 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
332, 19, 15, 32syl3anc 1323 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) 𝑄) ∈ 𝐵)
344, 9, 10latlej1 16976 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑋) ∈ 𝐵𝑄𝐵) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
352, 19, 15, 34syl3anc 1323 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → (𝑃 𝑋) ((𝑃 𝑋) 𝑄))
364, 9, 2, 7, 19, 33, 12, 35lattrd 16974 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → 𝑃 ((𝑃 𝑋) 𝑄))
374, 9, 10, 23, 5cvrat3 34194 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑃 𝑋) ∈ 𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 (𝑃 𝑋) ∧ 𝑃 ((𝑃 𝑋) 𝑄)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴))
38373impia 1258 . . . . 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 1346 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴)
409, 5atcmp 34064 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴 ∧ ((𝑃 𝑋) (𝑃 𝑄)) ∈ 𝐴) → (𝑃 ((𝑃 𝑋) (𝑃 𝑄)) ↔ 𝑃 = ((𝑃 𝑋) (𝑃 𝑄))))
4128, 3, 39, 40syl3anc 1323 . . 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 2632 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 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796   class class class wbr 4618  cfv 5850  (class class class)co 6605  Basecbs 15776  lecple 15864  joincjn 16860  meetcmee 16861  Latclat 16961  Atomscatm 34016  AtLatcal 34017  HLchlt 34103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-preset 16844  df-poset 16862  df-plt 16874  df-lub 16890  df-glb 16891  df-join 16892  df-meet 16893  df-p0 16955  df-lat 16962  df-clat 17024  df-oposet 33929  df-ol 33931  df-oml 33932  df-covers 34019  df-ats 34020  df-atl 34051  df-cvlat 34075  df-hlat 34104
This theorem is referenced by:  2llnma1  34539  cdlemg4  35371  cdlemkfid1N  35675
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