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Theorem cdleme13 36050
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, "<s,t,p> and <f(s),f(t),q> are centrally perspective." 𝐹 and 𝐺 represent f(s) and f(t) respectively. (Contributed by NM, 7-Oct-2012.)
Hypotheses
Ref Expression
cdleme12.l = (le‘𝐾)
cdleme12.j = (join‘𝐾)
cdleme12.m = (meet‘𝐾)
cdleme12.a 𝐴 = (Atoms‘𝐾)
cdleme12.h 𝐻 = (LHyp‘𝐾)
cdleme12.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme12.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme12.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
Assertion
Ref Expression
cdleme13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝑆 𝐹) (𝑇 𝐺)) (𝑃 𝑄))

Proof of Theorem cdleme13
StepHypRef Expression
1 cdleme12.l . . . 4 = (le‘𝐾)
2 cdleme12.j . . . 4 = (join‘𝐾)
3 cdleme12.m . . . 4 = (meet‘𝐾)
4 cdleme12.a . . . 4 𝐴 = (Atoms‘𝐾)
5 cdleme12.h . . . 4 𝐻 = (LHyp‘𝐾)
6 cdleme12.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
7 cdleme12.f . . . 4 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
8 cdleme12.g . . . 4 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
91, 2, 3, 4, 5, 6, 7, 8cdleme12 36049 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝑆 𝐹) (𝑇 𝐺)) = 𝑈)
109, 6syl6eq 2855 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝑆 𝐹) (𝑇 𝐺)) = ((𝑃 𝑄) 𝑊))
11 simp1l 1247 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝐾 ∈ HL)
1211hllatd 35141 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝐾 ∈ Lat)
13 simp21l 1382 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝑃𝐴)
14 simp22 1257 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝑄𝐴)
15 eqid 2805 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
1615, 2, 4hlatjcl 35144 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
1711, 13, 14, 16syl3anc 1483 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → (𝑃 𝑄) ∈ (Base‘𝐾))
18 simp1r 1248 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝑊𝐻)
1915, 5lhpbase 35775 . . . 4 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2018, 19syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → 𝑊 ∈ (Base‘𝐾))
2115, 1, 3latmle1 17277 . . 3 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
2212, 17, 20, 21syl3anc 1483 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝑃 𝑄) 𝑊) (𝑃 𝑄))
2310, 22eqbrtrd 4862 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝑇 ∧ ¬ 𝑈 (𝑆 𝑇)))) → ((𝑆 𝐹) (𝑇 𝐺)) (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2158  wne 2977   class class class wbr 4840  cfv 6098  (class class class)co 6871  Basecbs 16064  lecple 16156  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35040  HLchlt 35127  LHypclh 35761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-iin 4711  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-1st 7395  df-2nd 7396  df-proset 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 34953  df-ol 34955  df-oml 34956  df-covers 35043  df-ats 35044  df-atl 35075  df-cvlat 35099  df-hlat 35128  df-psubsp 35280  df-pmap 35281  df-padd 35573  df-lhyp 35765
This theorem is referenced by:  cdleme14  36051
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