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Theorem cdlemn2 35305
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 1053 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1086 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3 simp22 1087 . . . . . 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 34688 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → 𝐹𝑇)
101, 2, 3, 9syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐹𝑇)
11 cdlemn2.j . . . . . 6 = (join‘𝐾)
12 eqid 2609 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
13 cdlemn2.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
144, 11, 12, 5, 6, 7, 13trlval2 34271 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄))(meet‘𝐾)𝑊))
151, 10, 2, 14syl3anc 1317 . . . 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 34690 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → (𝐹𝑄) = 𝑆)
171, 2, 3, 16syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝐹𝑄) = 𝑆)
1817oveq2d 6543 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 (𝐹𝑄)) = (𝑄 𝑆))
1918oveq1d 6542 . . . 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 2643 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
21 simp1l 1077 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐾 ∈ HL)
22 hllat 33471 . . . . . . 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 1170 . . . . . . 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 33397 . . . . . . 7 (𝑄𝐴𝑄𝐵)
2724, 26syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄𝐵)
28 simp23l 1174 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑋𝐵)
2925, 4, 11latlej1 16829 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → 𝑄 (𝑄 𝑋))
3023, 27, 28, 29syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄 (𝑄 𝑋))
31 simp3 1055 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆 (𝑄 𝑋))
32 simp22l 1172 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆𝐴)
3325, 5atbase 33397 . . . . . . 7 (𝑆𝐴𝑆𝐵)
3432, 33syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆𝐵)
3525, 11latjcl 16820 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) ∈ 𝐵)
3623, 27, 28, 35syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑋) ∈ 𝐵)
3725, 4, 11latjle12 16831 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑆𝐵 ∧ (𝑄 𝑋) ∈ 𝐵)) → ((𝑄 (𝑄 𝑋) ∧ 𝑆 (𝑄 𝑋)) ↔ (𝑄 𝑆) (𝑄 𝑋)))
3823, 27, 34, 36, 37syl13anc 1319 . . . . 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 957 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑆) (𝑄 𝑋))
4025, 11, 5hlatjcl 33474 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ 𝐵)
4121, 24, 32, 40syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑆) ∈ 𝐵)
42 simp1r 1078 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑊𝐻)
4325, 6lhpbase 34105 . . . . . 6 (𝑊𝐻𝑊𝐵)
4442, 43syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑊𝐵)
4525, 4, 12latmlem1 16850 . . . . 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 1319 . . . 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 4599 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) ((𝑄 𝑋)(meet‘𝐾)𝑊))
49 simp23 1088 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑋𝐵𝑋 𝑊))
5025, 4, 11, 12, 5, 6lhple 34149 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑄 𝑋)(meet‘𝐾)𝑊) = 𝑋)
511, 2, 49, 50syl3anc 1317 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 𝑋)(meet‘𝐾)𝑊) = 𝑋)
5248, 51breqtrd 4603 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5790  crio 6488  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  Latclat 16814  Atomscatm 33371  HLchlt 33458  LHypclh 34091  LTrncltrn 34208  trLctrl 34266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-riotaBAD 33060
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-undef 7263  df-map 7723  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-llines 33605  df-lplanes 33606  df-lvols 33607  df-lines 33608  df-psubsp 33610  df-pmap 33611  df-padd 33903  df-lhyp 34095  df-laut 34096  df-ldil 34211  df-ltrn 34212  df-trl 34267
This theorem is referenced by:  cdlemn2a  35306
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