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Theorem cdleme15c 36351
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((p s1) (q s1)) ((p t1) (q t1))=s1 t1. 𝐶 and 𝑋 represent s1 and t1 respectively. The order of our operations is slightly different. (Contributed by NM, 10-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)))
cdleme15.c 𝐶 = ((𝑃 𝑆) 𝑊)
cdleme15.x 𝑋 = ((𝑃 𝑇) 𝑊)
Assertion
Ref Expression
cdleme15c ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (((𝑃 𝑋) (𝑄 𝑋)) ((𝑃 𝐶) (𝑄 𝐶))) = (𝑋 𝐶))

Proof of Theorem cdleme15c
StepHypRef Expression
1 simp11 1266 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp12 1267 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp13 1268 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp22 1270 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
5 simp21 1269 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
6 simp23l 1399 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝑃𝑄)
7 simp23r 1400 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝑆𝑇)
87necomd 3054 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝑇𝑆)
96, 8jca 509 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑃𝑄𝑇𝑆))
10 simp32 1273 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ¬ 𝑇 (𝑃 𝑄))
11 simp31 1272 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ¬ 𝑆 (𝑃 𝑄))
12 simp33 1274 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ¬ 𝑈 (𝑆 𝑇))
13 simp11l 1389 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝐾 ∈ HL)
14 simp21l 1395 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝑆𝐴)
15 simp22l 1397 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → 𝑇𝐴)
16 cdleme12.j . . . . . . 7 = (join‘𝐾)
17 cdleme12.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
1816, 17hlatjcom 35443 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) = (𝑇 𝑆))
1913, 14, 15, 18syl3anc 1496 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑆 𝑇) = (𝑇 𝑆))
2019breq2d 4885 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (𝑈 (𝑆 𝑇) ↔ 𝑈 (𝑇 𝑆)))
2112, 20mtbid 316 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ¬ 𝑈 (𝑇 𝑆))
22 cdleme12.l . . . 4 = (le‘𝐾)
23 cdleme12.m . . . 4 = (meet‘𝐾)
24 cdleme12.h . . . 4 𝐻 = (LHyp‘𝐾)
25 cdleme12.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
26 cdleme12.g . . . 4 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
27 cdleme12.f . . . 4 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
28 cdleme15.x . . . 4 𝑋 = ((𝑃 𝑇) 𝑊)
29 cdleme15.c . . . 4 𝐶 = ((𝑃 𝑆) 𝑊)
3022, 16, 23, 17, 24, 25, 26, 27, 28, 29cdleme15b 36350 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑃𝑄𝑇𝑆)) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑇 𝑆))) → ((𝑃 𝑋) (𝑄 𝑋)) = 𝑋)
311, 2, 3, 4, 5, 9, 10, 11, 21, 30syl333anc 1527 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ((𝑃 𝑋) (𝑄 𝑋)) = 𝑋)
3222, 16, 23, 17, 24, 25, 27, 26, 29, 28cdleme15b 36350 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → ((𝑃 𝐶) (𝑄 𝐶)) = 𝐶)
3331, 32oveq12d 6923 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑈 (𝑆 𝑇))) → (((𝑃 𝑋) (𝑄 𝑋)) ((𝑃 𝐶) (𝑄 𝐶))) = (𝑋 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2999   class class class wbr 4873  cfv 6123  (class class class)co 6905  lecple 16312  joincjn 17297  meetcmee 17298  Atomscatm 35338  HLchlt 35425  LHypclh 36059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35251  df-ol 35253  df-oml 35254  df-covers 35341  df-ats 35342  df-atl 35373  df-cvlat 35397  df-hlat 35426  df-psubsp 35578  df-pmap 35579  df-padd 35871  df-lhyp 36063
This theorem is referenced by:  cdleme15  36353
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