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Theorem cdlemn2 35502
Description: Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)
Hypotheses
Ref Expression
cdlemn2.b 𝐵 = (Base‘𝐾)
cdlemn2.l = (le‘𝐾)
cdlemn2.j = (join‘𝐾)
cdlemn2.a 𝐴 = (Atoms‘𝐾)
cdlemn2.h 𝐻 = (LHyp‘𝐾)
cdlemn2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn2.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemn2.f 𝐹 = (𝑇 (𝑄) = 𝑆)
Assertion
Ref Expression
cdlemn2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) 𝑋)
Distinct variable groups:   ,   𝐴,   ,𝐻   ,𝐾   𝑄,   𝑆,   𝑇,   ,𝑊
Allowed substitution hints:   𝐵()   𝑅()   𝐹()   ()   𝑋()

Proof of Theorem cdlemn2
StepHypRef Expression
1 simp1 1054 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1087 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3 simp22 1088 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
4 cdlemn2.l . . . . . . 7 = (le‘𝐾)
5 cdlemn2.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
6 cdlemn2.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
7 cdlemn2.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 cdlemn2.f . . . . . . 7 𝐹 = (𝑇 (𝑄) = 𝑆)
94, 5, 6, 7, 8ltrniotacl 34885 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → 𝐹𝑇)
101, 2, 3, 9syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐹𝑇)
11 cdlemn2.j . . . . . 6 = (join‘𝐾)
12 eqid 2610 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
13 cdlemn2.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
144, 11, 12, 5, 6, 7, 13trlval2 34468 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄))(meet‘𝐾)𝑊))
151, 10, 2, 14syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄))(meet‘𝐾)𝑊))
164, 5, 6, 7, 8ltrniotaval 34887 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → (𝐹𝑄) = 𝑆)
171, 2, 3, 16syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝐹𝑄) = 𝑆)
1817oveq2d 6565 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 (𝐹𝑄)) = (𝑄 𝑆))
1918oveq1d 6564 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 (𝐹𝑄))(meet‘𝐾)𝑊) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
2015, 19eqtrd 2644 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
21 simp1l 1078 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐾 ∈ HL)
22 hllat 33668 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2321, 22syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐾 ∈ Lat)
24 simp21l 1171 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄𝐴)
25 cdlemn2.b . . . . . . . 8 𝐵 = (Base‘𝐾)
2625, 5atbase 33594 . . . . . . 7 (𝑄𝐴𝑄𝐵)
2724, 26syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄𝐵)
28 simp23l 1175 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑋𝐵)
2925, 4, 11latlej1 16883 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → 𝑄 (𝑄 𝑋))
3023, 27, 28, 29syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄 (𝑄 𝑋))
31 simp3 1056 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆 (𝑄 𝑋))
32 simp22l 1173 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆𝐴)
3325, 5atbase 33594 . . . . . . 7 (𝑆𝐴𝑆𝐵)
3432, 33syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆𝐵)
3525, 11latjcl 16874 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) ∈ 𝐵)
3623, 27, 28, 35syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑋) ∈ 𝐵)
3725, 4, 11latjle12 16885 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑆𝐵 ∧ (𝑄 𝑋) ∈ 𝐵)) → ((𝑄 (𝑄 𝑋) ∧ 𝑆 (𝑄 𝑋)) ↔ (𝑄 𝑆) (𝑄 𝑋)))
3823, 27, 34, 36, 37syl13anc 1320 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 (𝑄 𝑋) ∧ 𝑆 (𝑄 𝑋)) ↔ (𝑄 𝑆) (𝑄 𝑋)))
3930, 31, 38mpbi2and 958 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑆) (𝑄 𝑋))
4025, 11, 5hlatjcl 33671 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ 𝐵)
4121, 24, 32, 40syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑆) ∈ 𝐵)
42 simp1r 1079 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑊𝐻)
4325, 6lhpbase 34302 . . . . . 6 (𝑊𝐻𝑊𝐵)
4442, 43syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑊𝐵)
4525, 4, 12latmlem1 16904 . . . . 5 ((𝐾 ∈ Lat ∧ ((𝑄 𝑆) ∈ 𝐵 ∧ (𝑄 𝑋) ∈ 𝐵𝑊𝐵)) → ((𝑄 𝑆) (𝑄 𝑋) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ((𝑄 𝑋)(meet‘𝐾)𝑊)))
4623, 41, 36, 44, 45syl13anc 1320 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 𝑆) (𝑄 𝑋) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ((𝑄 𝑋)(meet‘𝐾)𝑊)))
4739, 46mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ((𝑄 𝑋)(meet‘𝐾)𝑊))
4820, 47eqbrtrd 4605 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) ((𝑄 𝑋)(meet‘𝐾)𝑊))
49 simp23 1089 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑋𝐵𝑋 𝑊))
5025, 4, 11, 12, 5, 6lhple 34346 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑄 𝑋)(meet‘𝐾)𝑊) = 𝑋)
511, 2, 49, 50syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 𝑋)(meet‘𝐾)𝑊) = 𝑋)
5248, 51breqtrd 4609 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583  cfv 5804  crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464
This theorem is referenced by:  cdlemn2a  35503
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