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Theorem cdleme20bN 35915
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). We show v s2 = v t2. (Contributed by NM, 15-Nov-2012.) (New usage is discouraged.)
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)
cdleme20.v 𝑉 = ((𝑆 𝑇) 𝑊)
Assertion
Ref Expression
cdleme20bN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (𝑉 𝑌))

Proof of Theorem cdleme20bN
StepHypRef Expression
1 simp1l 1105 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
2 hllat 34968 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ Lat)
4 simp22l 1200 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑆𝐴)
5 eqid 2651 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
6 cdleme19.a . . . . . 6 𝐴 = (Atoms‘𝐾)
75, 6atbase 34894 . . . . 5 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
84, 7syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑆 ∈ (Base‘𝐾))
9 simp21 1114 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅𝐴)
105, 6atbase 34894 . . . . 5 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
119, 10syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 ∈ (Base‘𝐾))
12 simp23l 1202 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑇𝐴)
135, 6atbase 34894 . . . . 5 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1412, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑇 ∈ (Base‘𝐾))
15 cdleme19.j . . . . 5 = (join‘𝐾)
165, 15latj31 17146 . . . 4 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾))) → ((𝑆 𝑅) 𝑇) = ((𝑇 𝑅) 𝑆))
173, 8, 11, 14, 16syl13anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝑆 𝑅) 𝑇) = ((𝑇 𝑅) 𝑆))
1817oveq1d 6705 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (((𝑆 𝑅) 𝑇) 𝑊) = (((𝑇 𝑅) 𝑆) 𝑊))
19 simp1r 1106 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑊𝐻)
20 simp22r 1201 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑆 𝑊)
21 simp31 1117 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
22 simp33 1119 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
23 cdleme19.l . . . 4 = (le‘𝐾)
24 cdleme19.m . . . 4 = (meet‘𝐾)
25 cdleme19.h . . . 4 𝐻 = (LHyp‘𝐾)
26 cdleme19.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
27 cdleme19.f . . . 4 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
28 cdleme19.g . . . 4 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
29 cdleme19.d . . . 4 𝐷 = ((𝑅 𝑆) 𝑊)
30 cdleme19.y . . . 4 𝑌 = ((𝑅 𝑇) 𝑊)
31 cdleme20.v . . . 4 𝑉 = ((𝑆 𝑇) 𝑊)
3223, 15, 24, 6, 25, 26, 27, 28, 29, 30, 31cdleme20aN 35914 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (((𝑆 𝑅) 𝑇) 𝑊))
331, 19, 9, 4, 20, 12, 21, 22, 32syl233anc 1395 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (((𝑆 𝑅) 𝑇) 𝑊))
3415, 6hlatjcom 34972 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
351, 4, 12, 34syl3anc 1366 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 𝑇) = (𝑇 𝑆))
3635oveq1d 6705 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝑆 𝑇) 𝑊) = ((𝑇 𝑆) 𝑊))
3731, 36syl5eq 2697 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑉 = ((𝑇 𝑆) 𝑊))
3837oveq1d 6705 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝑌) = (((𝑇 𝑆) 𝑊) 𝑌))
39 simp23r 1203 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑇 𝑊)
40 simp32 1118 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑇 (𝑃 𝑄))
41 eqid 2651 . . . . 5 ((𝑇 𝑆) 𝑊) = ((𝑇 𝑆) 𝑊)
4223, 15, 24, 6, 25, 26, 28, 27, 30, 29, 41cdleme20aN 35914 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (((𝑇 𝑆) 𝑊) 𝑌) = (((𝑇 𝑅) 𝑆) 𝑊))
431, 19, 9, 12, 39, 4, 40, 22, 42syl233anc 1395 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (((𝑇 𝑆) 𝑊) 𝑌) = (((𝑇 𝑅) 𝑆) 𝑊))
4438, 43eqtrd 2685 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝑌) = (((𝑇 𝑅) 𝑆) 𝑊))
4518, 33, 443eqtr4d 2695 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (𝑉 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Latclat 17092  Atomscatm 34868  HLchlt 34955  LHypclh 35588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592
This theorem is referenced by: (None)
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