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Theorem cdleme19a 36078
Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷 represents s2. In their notation, we prove that if r s t, then s2=(s t) w. (Contributed by NM, 13-Nov-2012.)
Hypotheses
Ref Expression
cdleme19.l = (le‘𝐾)
cdleme19.j = (join‘𝐾)
cdleme19.m = (meet‘𝐾)
cdleme19.a 𝐴 = (Atoms‘𝐾)
cdleme19.h 𝐻 = (LHyp‘𝐾)
cdleme19.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme19.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme19.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme19.d 𝐷 = ((𝑅 𝑆) 𝑊)
cdleme19.y 𝑌 = ((𝑅 𝑇) 𝑊)
Assertion
Ref Expression
cdleme19a ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐷 = ((𝑆 𝑇) 𝑊))

Proof of Theorem cdleme19a
StepHypRef Expression
1 cdleme19.d . 2 𝐷 = ((𝑅 𝑆) 𝑊)
2 eqid 2806 . . . 4 (Base‘𝐾) = (Base‘𝐾)
3 cdleme19.l . . . 4 = (le‘𝐾)
4 hllat 35138 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
543ad2ant1 1156 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ Lat)
6 simp1 1159 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ HL)
7 simp21 1256 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅𝐴)
8 simp22 1257 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑆𝐴)
9 cdleme19.j . . . . . 6 = (join‘𝐾)
10 cdleme19.a . . . . . 6 𝐴 = (Atoms‘𝐾)
112, 9, 10hlatjcl 35142 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
126, 7, 8, 11syl3anc 1483 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) ∈ (Base‘𝐾))
13 simp23 1258 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑇𝐴)
142, 9, 10hlatjcl 35142 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
156, 8, 13, 14syl3anc 1483 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑆 𝑇) ∈ (Base‘𝐾))
16 simp33 1261 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅 (𝑆 𝑇))
173, 9, 10hlatlej1 35150 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → 𝑆 (𝑆 𝑇))
186, 8, 13, 17syl3anc 1483 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑆 (𝑆 𝑇))
192, 10atbase 35064 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
207, 19syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅 ∈ (Base‘𝐾))
212, 10atbase 35064 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
228, 21syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑆 ∈ (Base‘𝐾))
232, 3, 9latjle12 17263 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑆 𝑇) ∈ (Base‘𝐾))) → ((𝑅 (𝑆 𝑇) ∧ 𝑆 (𝑆 𝑇)) ↔ (𝑅 𝑆) (𝑆 𝑇)))
245, 20, 22, 15, 23syl13anc 1484 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → ((𝑅 (𝑆 𝑇) ∧ 𝑆 (𝑆 𝑇)) ↔ (𝑅 𝑆) (𝑆 𝑇)))
2516, 18, 24mpbi2and 694 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) (𝑆 𝑇))
263, 9, 10hlatlej2 35151 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑆 (𝑅 𝑆))
276, 7, 8, 26syl3anc 1483 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑆 (𝑅 𝑆))
28 hlcvl 35134 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
29283ad2ant1 1156 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ CvLat)
30 simp31 1259 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅 (𝑃 𝑄))
31 simp32 1260 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → ¬ 𝑆 (𝑃 𝑄))
32 nbrne2 4864 . . . . . . . . 9 ((𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑅𝑆)
3330, 31, 32syl2anc 575 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅𝑆)
343, 9, 10cvlatexch1 35111 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑅𝐴𝑇𝐴𝑆𝐴) ∧ 𝑅𝑆) → (𝑅 (𝑆 𝑇) → 𝑇 (𝑆 𝑅)))
3529, 7, 13, 8, 33, 34syl131anc 1495 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 (𝑆 𝑇) → 𝑇 (𝑆 𝑅)))
3616, 35mpd 15 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑇 (𝑆 𝑅))
379, 10hlatjcom 35143 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) = (𝑆 𝑅))
386, 7, 8, 37syl3anc 1483 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) = (𝑆 𝑅))
3936, 38breqtrrd 4872 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑇 (𝑅 𝑆))
402, 10atbase 35064 . . . . . . 7 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
4113, 40syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑇 ∈ (Base‘𝐾))
422, 3, 9latjle12 17263 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾))) → ((𝑆 (𝑅 𝑆) ∧ 𝑇 (𝑅 𝑆)) ↔ (𝑆 𝑇) (𝑅 𝑆)))
435, 22, 41, 12, 42syl13anc 1484 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → ((𝑆 (𝑅 𝑆) ∧ 𝑇 (𝑅 𝑆)) ↔ (𝑆 𝑇) (𝑅 𝑆)))
4427, 39, 43mpbi2and 694 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑆 𝑇) (𝑅 𝑆))
452, 3, 5, 12, 15, 25, 44latasymd 17258 . . 3 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) = (𝑆 𝑇))
4645oveq1d 6885 . 2 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → ((𝑅 𝑆) 𝑊) = ((𝑆 𝑇) 𝑊))
471, 46syl5eq 2852 1 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐷 = ((𝑆 𝑇) 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978   class class class wbr 4844  cfv 6097  (class class class)co 6870  Basecbs 16064  lecple 16156  joincjn 17145  meetcmee 17146  Latclat 17246  Atomscatm 35038  CvLatclc 35040  HLchlt 35125  LHypclh 35759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
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 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  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-lat 17247  df-covers 35041  df-ats 35042  df-atl 35073  df-cvlat 35097  df-hlat 35126
This theorem is referenced by:  cdleme19b  36079
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