Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme19b Structured version   Visualization version   GIF version

Theorem cdleme19b 37559
Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line. 𝐷, 𝐹, 𝐺 represent s2, f(s), f(t). In their notation, we prove that if r s t, then s2 f(s) f(t). (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
cdleme19b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷 (𝐹 𝐺))

Proof of Theorem cdleme19b
StepHypRef Expression
1 simp11l 1281 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐾 ∈ HL)
2 simp23 1205 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑅𝐴)
3 simp21l 1287 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑆𝐴)
4 simp22l 1289 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑇𝐴)
5 simp33l 1297 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑅 (𝑃 𝑄))
6 simp32l 1295 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ¬ 𝑆 (𝑃 𝑄))
7 simp33r 1298 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑅 (𝑆 𝑇))
8 cdleme19.l . . . . 5 = (le‘𝐾)
9 cdleme19.j . . . . 5 = (join‘𝐾)
10 cdleme19.m . . . . 5 = (meet‘𝐾)
11 cdleme19.a . . . . 5 𝐴 = (Atoms‘𝐾)
12 cdleme19.h . . . . 5 𝐻 = (LHyp‘𝐾)
13 cdleme19.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
14 cdleme19.f . . . . 5 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
15 cdleme19.g . . . . 5 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
16 cdleme19.d . . . . 5 𝐷 = ((𝑅 𝑆) 𝑊)
17 cdleme19.y . . . . 5 𝑌 = ((𝑅 𝑇) 𝑊)
188, 9, 10, 11, 12, 13, 14, 15, 16, 17cdleme19a 37558 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑆𝐴𝑇𝐴) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇))) → 𝐷 = ((𝑆 𝑇) 𝑊))
191, 2, 3, 4, 5, 6, 7, 18syl133anc 1390 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷 = ((𝑆 𝑇) 𝑊))
20 simp11 1200 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
21 simp12 1201 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
22 simp13 1202 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
23 simp21 1203 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
24 simp22 1204 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
25 simp31 1206 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝑃𝑄𝑆𝑇))
26 simp32r 1296 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ¬ 𝑇 (𝑃 𝑄))
278, 9, 10, 11, 12, 13, 14, 15cdleme16 37540 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) → ((𝑆 𝑇) 𝑊) = ((𝐹 𝐺) 𝑊))
2820, 21, 22, 23, 24, 25, 6, 26, 27syl332anc 1398 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ((𝑆 𝑇) 𝑊) = ((𝐹 𝐺) 𝑊))
2919, 28eqtrd 2857 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷 = ((𝐹 𝐺) 𝑊))
301hllatd 36619 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐾 ∈ Lat)
31 simp11r 1282 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑊𝐻)
32 simp12l 1283 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑃𝐴)
33 simp13l 1285 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑄𝐴)
34 eqid 2822 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
358, 9, 10, 11, 12, 13, 14, 34cdleme1b 37481 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → 𝐹 ∈ (Base‘𝐾))
361, 31, 32, 33, 3, 35syl23anc 1374 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐹 ∈ (Base‘𝐾))
378, 9, 10, 11, 12, 13, 15, 34cdleme1b 37481 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑇𝐴)) → 𝐺 ∈ (Base‘𝐾))
381, 31, 32, 33, 4, 37syl23anc 1374 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐺 ∈ (Base‘𝐾))
3934, 9latjcl 17652 . . . 4 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾)) → (𝐹 𝐺) ∈ (Base‘𝐾))
4030, 36, 38, 39syl3anc 1368 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → (𝐹 𝐺) ∈ (Base‘𝐾))
4134, 12lhpbase 37253 . . . 4 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
4231, 41syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝑊 ∈ (Base‘𝐾))
4334, 8, 10latmle1 17677 . . 3 ((𝐾 ∈ Lat ∧ (𝐹 𝐺) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝐹 𝐺) 𝑊) (𝐹 𝐺))
4430, 40, 42, 43syl3anc 1368 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → ((𝐹 𝐺) 𝑊) (𝐹 𝐺))
4529, 44eqbrtrd 5064 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ 𝑅𝐴) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑅 (𝑆 𝑇)))) → 𝐷 (𝐹 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  lecple 16563  joincjn 17545  meetcmee 17546  Latclat 17646  Atomscatm 36518  HLchlt 36605  LHypclh 37239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36431  df-ol 36433  df-oml 36434  df-covers 36521  df-ats 36522  df-atl 36553  df-cvlat 36577  df-hlat 36606  df-llines 36753  df-lplanes 36754  df-lvols 36755  df-lines 36756  df-psubsp 36758  df-pmap 36759  df-padd 37051  df-lhyp 37243
This theorem is referenced by:  cdleme19d  37561
  Copyright terms: Public domain W3C validator