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Theorem cdleme23a 34454
Description: Part of proof of Lemma E in [Crawley] p. 113. (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
cdleme23a ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑉 𝑊)

Proof of Theorem cdleme23a
StepHypRef Expression
1 cdleme23.v . 2 𝑉 = ((𝑆 𝑇) (𝑋 𝑊))
2 cdleme23.b . . 3 𝐵 = (Base‘𝐾)
3 cdleme23.l . . 3 = (le‘𝐾)
4 simp11l 1164 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
5 hllat 33467 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
64, 5syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
7 simp12l 1166 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆𝐴)
8 simp13l 1168 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑇𝐴)
9 cdleme23.j . . . . . 6 = (join‘𝐾)
10 cdleme23.a . . . . . 6 𝐴 = (Atoms‘𝐾)
112, 9, 10hlatjcl 33470 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ 𝐵)
124, 7, 8, 11syl3anc 1317 . . . 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)
13 simp2l 1079 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
14 simp11r 1165 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
15 cdleme23.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
162, 15lhpbase 34101 . . . . . 6 (𝑊𝐻𝑊𝐵)
1714, 16syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
18 cdleme23.m . . . . . 6 = (meet‘𝐾)
192, 18latmcl 16817 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
206, 13, 17, 19syl3anc 1317 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) ∈ 𝐵)
212, 18latmcl 16817 . . . 4 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → ((𝑆 𝑇) (𝑋 𝑊)) ∈ 𝐵)
226, 12, 20, 21syl3anc 1317 . . 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)) ∈ 𝐵)
232, 3, 18latmle2 16842 . . . 4 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → ((𝑆 𝑇) (𝑋 𝑊)) (𝑋 𝑊))
246, 12, 20, 23syl3anc 1317 . . 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)) (𝑋 𝑊))
252, 3, 18latmle2 16842 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
266, 13, 17, 25syl3anc 1317 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) 𝑊)
272, 3, 6, 22, 20, 17, 24, 26lattrd 16823 . 2 ((((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)) 𝑊)
281, 27syl5eqbr 4608 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑉 𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  Latclat 16810  Atomscatm 33367  HLchlt 33454  LHypclh 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-poset 16711  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-lat 16811  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-lhyp 34091
This theorem is referenced by:  cdleme28a  34475
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