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Theorem cdleme19a 36970
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 2795 . . . 4 (Base‘𝐾) = (Base‘𝐾)
3 cdleme19.l . . . 4 = (le‘𝐾)
4 hllat 36030 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
543ad2ant1 1126 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ Lat)
6 simp1 1129 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ HL)
7 simp21 1199 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅𝐴)
8 simp22 1200 . . . . 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 36034 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
126, 7, 8, 11syl3anc 1364 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) ∈ (Base‘𝐾))
13 simp23 1201 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑇𝐴)
142, 9, 10hlatjcl 36034 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
156, 8, 13, 14syl3anc 1364 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑆 𝑇) ∈ (Base‘𝐾))
16 simp33 1204 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅 (𝑆 𝑇))
173, 9, 10hlatlej1 36042 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → 𝑆 (𝑆 𝑇))
186, 8, 13, 17syl3anc 1364 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑆 (𝑆 𝑇))
192, 10atbase 35956 . . . . . . 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 35956 . . . . . . 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 17501 . . . . . 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 1365 . . . . 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 708 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) (𝑆 𝑇))
263, 9, 10hlatlej2 36043 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑆 (𝑅 𝑆))
276, 7, 8, 26syl3anc 1364 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑆 (𝑅 𝑆))
28 hlcvl 36026 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
29283ad2ant1 1126 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ CvLat)
30 simp31 1202 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅 (𝑃 𝑄))
31 simp32 1203 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → ¬ 𝑆 (𝑃 𝑄))
32 nbrne2 4982 . . . . . . . . 9 ((𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑅𝑆)
3330, 31, 32syl2anc 584 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑅𝑆)
343, 9, 10cvlatexch1 36003 . . . . . . . 8 ((𝐾 ∈ CvLat ∧ (𝑅𝐴𝑇𝐴𝑆𝐴) ∧ 𝑅𝑆) → (𝑅 (𝑆 𝑇) → 𝑇 (𝑆 𝑅)))
3529, 7, 13, 8, 33, 34syl131anc 1376 . . . . . . 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 36035 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) = (𝑆 𝑅))
386, 7, 8, 37syl3anc 1364 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) = (𝑆 𝑅))
3936, 38breqtrrd 4990 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝑇 (𝑅 𝑆))
402, 10atbase 35956 . . . . . . 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 17501 . . . . . 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 1365 . . . . 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 708 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑆 𝑇) (𝑅 𝑆))
452, 3, 5, 12, 15, 25, 44latasymd 17496 . . 3 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → (𝑅 𝑆) = (𝑆 𝑇))
4645oveq1d 7031 . 2 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → ((𝑅 𝑆) 𝑊) = ((𝑆 𝑇) 𝑊))
471, 46syl5eq 2843 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984   class class class wbr 4962  cfv 6225  (class class class)co 7016  Basecbs 16312  lecple 16401  joincjn 17383  meetcmee 17384  Latclat 17484  Atomscatm 35930  CvLatclc 35932  HLchlt 36017  LHypclh 36651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-proset 17367  df-poset 17385  df-plt 17397  df-lub 17413  df-glb 17414  df-join 17415  df-meet 17416  df-p0 17478  df-lat 17485  df-covers 35933  df-ats 35934  df-atl 35965  df-cvlat 35989  df-hlat 36018
This theorem is referenced by:  cdleme19b  36971
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