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Theorem cdleme23c 40334
Description: Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
Hypotheses
Ref Expression
cdleme23.b 𝐵 = (Base‘𝐾)
cdleme23.l = (le‘𝐾)
cdleme23.j = (join‘𝐾)
cdleme23.m = (meet‘𝐾)
cdleme23.a 𝐴 = (Atoms‘𝐾)
cdleme23.h 𝐻 = (LHyp‘𝐾)
cdleme23.v 𝑉 = ((𝑆 𝑇) (𝑋 𝑊))
Assertion
Ref Expression
cdleme23c ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 (𝑇 𝑉))

Proof of Theorem cdleme23c
StepHypRef Expression
1 simp11l 1283 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
21hllatd 39346 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
3 simp12l 1285 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆𝐴)
4 cdleme23.b . . . . . 6 𝐵 = (Base‘𝐾)
5 cdleme23.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atbase 39271 . . . . 5 (𝑆𝐴𝑆𝐵)
73, 6syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆𝐵)
8 simp13l 1287 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑇𝐴)
94, 5atbase 39271 . . . . 5 (𝑇𝐴𝑇𝐵)
108, 9syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑇𝐵)
11 cdleme23.l . . . . 5 = (le‘𝐾)
12 cdleme23.j . . . . 5 = (join‘𝐾)
134, 11, 12latlej1 18506 . . . 4 ((𝐾 ∈ Lat ∧ 𝑆𝐵𝑇𝐵) → 𝑆 (𝑆 𝑇))
142, 7, 10, 13syl3anc 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 (𝑆 𝑇))
15 simp2l 1198 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
16 simp11r 1284 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
17 cdleme23.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
184, 17lhpbase 39981 . . . . . . 7 (𝑊𝐻𝑊𝐵)
1916, 18syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
20 cdleme23.m . . . . . . 7 = (meet‘𝐾)
214, 20latmcl 18498 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
222, 15, 19, 21syl3anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) ∈ 𝐵)
234, 11, 12latlej1 18506 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑆𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑆 (𝑆 (𝑋 𝑊)))
242, 7, 22, 23syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 (𝑆 (𝑋 𝑊)))
25 simp32 1209 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑆 (𝑋 𝑊)) = 𝑋)
26 simp33 1210 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑇 (𝑋 𝑊)) = 𝑋)
2725, 26eqtr4d 2778 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑆 (𝑋 𝑊)) = (𝑇 (𝑋 𝑊)))
2824, 27breqtrd 5174 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 (𝑇 (𝑋 𝑊)))
294, 12, 5hlatjcl 39349 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ 𝐵)
301, 3, 8, 29syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑆 𝑇) ∈ 𝐵)
314, 12latjcl 18497 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑇𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑇 (𝑋 𝑊)) ∈ 𝐵)
322, 10, 22, 31syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑇 (𝑋 𝑊)) ∈ 𝐵)
334, 11, 20latlem12 18524 . . . 4 ((𝐾 ∈ Lat ∧ (𝑆𝐵 ∧ (𝑆 𝑇) ∈ 𝐵 ∧ (𝑇 (𝑋 𝑊)) ∈ 𝐵)) → ((𝑆 (𝑆 𝑇) ∧ 𝑆 (𝑇 (𝑋 𝑊))) ↔ 𝑆 ((𝑆 𝑇) (𝑇 (𝑋 𝑊)))))
342, 7, 30, 32, 33syl13anc 1371 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 (𝑆 𝑇) ∧ 𝑆 (𝑇 (𝑋 𝑊))) ↔ 𝑆 ((𝑆 𝑇) (𝑇 (𝑋 𝑊)))))
3514, 28, 34mpbi2and 712 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 ((𝑆 𝑇) (𝑇 (𝑋 𝑊))))
36 cdleme23.v . . . 4 𝑉 = ((𝑆 𝑇) (𝑋 𝑊))
3736oveq2i 7442 . . 3 (𝑇 𝑉) = (𝑇 ((𝑆 𝑇) (𝑋 𝑊)))
384, 11, 12latlej2 18507 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑆𝐵𝑇𝐵) → 𝑇 (𝑆 𝑇))
392, 7, 10, 38syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑇 (𝑆 𝑇))
404, 11, 12, 20, 5atmod3i1 39847 . . . 4 ((𝐾 ∈ HL ∧ (𝑇𝐴 ∧ (𝑆 𝑇) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) ∧ 𝑇 (𝑆 𝑇)) → (𝑇 ((𝑆 𝑇) (𝑋 𝑊))) = ((𝑆 𝑇) (𝑇 (𝑋 𝑊))))
411, 8, 30, 22, 39, 40syl131anc 1382 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑇 ((𝑆 𝑇) (𝑋 𝑊))) = ((𝑆 𝑇) (𝑇 (𝑋 𝑊))))
4237, 41eqtrid 2787 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑇 𝑉) = ((𝑆 𝑇) (𝑇 (𝑋 𝑊))))
4335, 42breqtrrd 5176 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆 (𝑇 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  HLchlt 39332  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971
This theorem is referenced by:  cdleme28a  40353
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