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Theorem cdleme20yOLD 35056
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) Obsolete version of cdleme20y 35055 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cdleme20z.l = (le‘𝐾)
cdleme20z.j = (join‘𝐾)
cdleme20z.m = (meet‘𝐾)
cdleme20z.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cdleme20yOLD ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((𝑆 𝑅) (𝑇 𝑅)) = 𝑅)

Proof of Theorem cdleme20yOLD
StepHypRef Expression
1 simp3r 1088 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ¬ 𝑅 (𝑆 𝑇))
2 simp1 1059 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ HL)
3 simp22 1093 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝑆𝐴)
4 simp23 1094 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝑇𝐴)
5 cdleme20z.j . . . . . . . . 9 = (join‘𝐾)
6 cdleme20z.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
75, 6hlatjcom 34120 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
82, 3, 4, 7syl3anc 1323 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → (𝑆 𝑇) = (𝑇 𝑆))
98breq2d 4630 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → (𝑅 (𝑆 𝑇) ↔ 𝑅 (𝑇 𝑆)))
101, 9mtbid 314 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ¬ 𝑅 (𝑇 𝑆))
11 hlcvl 34112 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
12113ad2ant1 1080 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ CvLat)
13 simp21 1092 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝑅𝐴)
14 simp3l 1087 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝑆𝑇)
15 cdleme20z.l . . . . . . 7 = (le‘𝐾)
1615, 5, 6cvlatexch1 34089 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑆𝐴𝑅𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 (𝑇 𝑅) → 𝑅 (𝑇 𝑆)))
1712, 3, 13, 4, 14, 16syl131anc 1336 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → (𝑆 (𝑇 𝑅) → 𝑅 (𝑇 𝑆)))
1810, 17mtod 189 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ¬ 𝑆 (𝑇 𝑅))
19 hlatl 34113 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
20193ad2ant1 1080 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ AtLat)
21 eqid 2626 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2221, 5, 6hlatjcl 34119 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑅𝐴) → (𝑇 𝑅) ∈ (Base‘𝐾))
232, 4, 13, 22syl3anc 1323 . . . . 5 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → (𝑇 𝑅) ∈ (Base‘𝐾))
24 cdleme20z.m . . . . . 6 = (meet‘𝐾)
25 eqid 2626 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
2621, 15, 24, 25, 6atnle 34070 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑆𝐴 ∧ (𝑇 𝑅) ∈ (Base‘𝐾)) → (¬ 𝑆 (𝑇 𝑅) ↔ (𝑆 (𝑇 𝑅)) = (0.‘𝐾)))
2720, 3, 23, 26syl3anc 1323 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → (¬ 𝑆 (𝑇 𝑅) ↔ (𝑆 (𝑇 𝑅)) = (0.‘𝐾)))
2818, 27mpbid 222 . . 3 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → (𝑆 (𝑇 𝑅)) = (0.‘𝐾))
2928oveq1d 6620 . 2 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((𝑆 (𝑇 𝑅)) 𝑅) = ((0.‘𝐾) 𝑅))
3021, 6atbase 34042 . . . 4 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
3113, 30syl 17 . . 3 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝑅 ∈ (Base‘𝐾))
3215, 5, 6hlatlej2 34128 . . . 4 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑅𝐴) → 𝑅 (𝑇 𝑅))
332, 4, 13, 32syl3anc 1323 . . 3 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝑅 (𝑇 𝑅))
3421, 15, 5, 24, 6atmod4i2 34619 . . 3 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑅 ∈ (Base‘𝐾) ∧ (𝑇 𝑅) ∈ (Base‘𝐾)) ∧ 𝑅 (𝑇 𝑅)) → ((𝑆 (𝑇 𝑅)) 𝑅) = ((𝑆 𝑅) (𝑇 𝑅)))
352, 3, 31, 23, 33, 34syl131anc 1336 . 2 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((𝑆 (𝑇 𝑅)) 𝑅) = ((𝑆 𝑅) (𝑇 𝑅)))
36 hlol 34114 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ OL)
37363ad2ant1 1080 . . 3 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → 𝐾 ∈ OL)
3821, 5, 25olj02 33979 . . 3 ((𝐾 ∈ OL ∧ 𝑅 ∈ (Base‘𝐾)) → ((0.‘𝐾) 𝑅) = 𝑅)
3937, 31, 38syl2anc 692 . 2 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((0.‘𝐾) 𝑅) = 𝑅)
4029, 35, 393eqtr3d 2668 1 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑆𝑇 ∧ ¬ 𝑅 (𝑆 𝑇))) → ((𝑆 𝑅) (𝑇 𝑅)) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796   class class class wbr 4618  cfv 5850  (class class class)co 6605  Basecbs 15776  lecple 15864  joincjn 16860  meetcmee 16861  0.cp0 16953  OLcol 33927  Atomscatm 34016  AtLatcal 34017  CvLatclc 34018  HLchlt 34103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-preset 16844  df-poset 16862  df-plt 16874  df-lub 16890  df-glb 16891  df-join 16892  df-meet 16893  df-p0 16955  df-lat 16962  df-clat 17024  df-oposet 33929  df-ol 33931  df-oml 33932  df-covers 34019  df-ats 34020  df-atl 34051  df-cvlat 34075  df-hlat 34104  df-psubsp 34255  df-pmap 34256  df-padd 34548
This theorem is referenced by: (None)
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