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Theorem cdlemn2 36986
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 1131 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1249 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3 simp22 1250 . . . . . 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 36369 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → 𝐹𝑇)
101, 2, 3, 9syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐹𝑇)
11 cdlemn2.j . . . . . 6 = (join‘𝐾)
12 eqid 2760 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
13 cdlemn2.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
144, 11, 12, 5, 6, 7, 13trlval2 35953 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝐹) = ((𝑄 (𝐹𝑄))(meet‘𝐾)𝑊))
151, 10, 2, 14syl3anc 1477 . . . 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 36371 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → (𝐹𝑄) = 𝑆)
171, 2, 3, 16syl3anc 1477 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝐹𝑄) = 𝑆)
1817oveq2d 6829 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 (𝐹𝑄)) = (𝑄 𝑆))
1918oveq1d 6828 . . . 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 2794 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
21 simp1l 1240 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝐾 ∈ HL)
22 hllat 35153 . . . . . . 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 1375 . . . . . . 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 35079 . . . . . . 7 (𝑄𝐴𝑄𝐵)
2724, 26syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄𝐵)
28 simp23l 1379 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑋𝐵)
2925, 4, 11latlej1 17261 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → 𝑄 (𝑄 𝑋))
3023, 27, 28, 29syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑄 (𝑄 𝑋))
31 simp3 1133 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆 (𝑄 𝑋))
32 simp22l 1377 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆𝐴)
3325, 5atbase 35079 . . . . . . 7 (𝑆𝐴𝑆𝐵)
3432, 33syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑆𝐵)
3525, 11latjcl 17252 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) ∈ 𝐵)
3623, 27, 28, 35syl3anc 1477 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑋) ∈ 𝐵)
3725, 4, 11latjle12 17263 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄𝐵𝑆𝐵 ∧ (𝑄 𝑋) ∈ 𝐵)) → ((𝑄 (𝑄 𝑋) ∧ 𝑆 (𝑄 𝑋)) ↔ (𝑄 𝑆) (𝑄 𝑋)))
3823, 27, 34, 36, 37syl13anc 1479 . . . . 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 994 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑆) (𝑄 𝑋))
4025, 11, 5hlatjcl 35156 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ 𝐵)
4121, 24, 32, 40syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑄 𝑆) ∈ 𝐵)
42 simp1r 1241 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑊𝐻)
4325, 6lhpbase 35787 . . . . . 6 (𝑊𝐻𝑊𝐵)
4442, 43syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → 𝑊𝐵)
4525, 4, 12latmlem1 17282 . . . . 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 1479 . . . 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 4826 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) ((𝑄 𝑋)(meet‘𝐾)𝑊))
49 simp23 1251 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑋𝐵𝑋 𝑊))
5025, 4, 11, 12, 5, 6lhple 35831 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝑄 𝑋)(meet‘𝐾)𝑊) = 𝑋)
511, 2, 49, 50syl3anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → ((𝑄 𝑋)(meet‘𝐾)𝑊) = 𝑋)
5248, 51breqtrd 4830 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  crio 6773  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35053  HLchlt 35140  LHypclh 35773  LTrncltrn 35890  trLctrl 35948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-riotaBAD 34742
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-undef 7568  df-map 8025  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288  df-lvols 35289  df-lines 35290  df-psubsp 35292  df-pmap 35293  df-padd 35585  df-lhyp 35777  df-laut 35778  df-ldil 35893  df-ltrn 35894  df-trl 35949
This theorem is referenced by:  cdlemn2a  36987
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