Theorem cdlemc5 41345

Description: Lemma for cdlemc 41347. (Contributed by NM, 26-May-2012.)

Hypotheses

Ref	Expression
cdlemc3.l	⊢ ≤ = (le‘𝐾)
cdlemc3.j	⊢ ∨ = (join‘𝐾)
cdlemc3.m	⊢ ∧ = (meet‘𝐾)
cdlemc3.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemc3.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemc3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemc3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemc5	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))

Proof of Theorem cdlemc5

Step	Hyp	Ref	Expression
1		simp1l 1216	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ HL)
2		simp23l 1313	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ∈ 𝐴)
3		simp1 1154	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		simp21 1225	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐹 ∈ 𝑇)
5		cdlemc3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6		cdlemc3.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemc3.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemc3.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	5, 6, 7, 8	ltrnat 41290	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
10	3, 4, 2, 9	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ∈ 𝐴)
11		cdlemc3.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
12	5, 11, 6	hlatlej2 40526	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝐹‘𝑄)))
13	1, 2, 10, 12	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝐹‘𝑄)))
14		simp23 1227	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
15		cdlemc3.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
16	5, 11, 6, 7, 8, 15	trljat1 41316	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (𝐹‘𝑄)))
17	3, 4, 14, 16	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (𝐹‘𝑄)))
18	13, 17	breqtrrd 5132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝑅‘𝐹)))
19		simp22 1226	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
20		cdlemc3.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
21	5, 11, 20, 6, 7, 8	cdlemc2 41342	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
22	3, 4, 19, 14, 21	syl112anc 1401	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
23	1	hllatd 40514	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ Lat)
24		eqid 2760	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 6	atbase 40439	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
26	2, 25	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ∈ (Base‘𝐾))
27	24, 7, 8	ltrncl 41275	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐹‘𝑄) ∈ (Base‘𝐾))
28	3, 4, 26, 27	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ∈ (Base‘𝐾))
29	24, 7, 8, 15	trlcl 41314	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	3, 4, 29	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ (Base‘𝐾))
31	24, 11	latjcl 18618	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾))
32	23, 26, 30, 31	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾))
33		simp22l 1311	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ∈ 𝐴)
34	24, 6	atbase 40439	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
35	33, 34	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ∈ (Base‘𝐾))
36	24, 7, 8	ltrncl 41275	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
37	3, 4, 35, 36	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
38	24, 11, 6	hlatjcl 40517	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
39	1, 33, 2, 38	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
40		simp1r 1217	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑊 ∈ 𝐻)
41	24, 7	lhpbase 41148	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
42	40, 41	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑊 ∈ (Base‘𝐾))
43	24, 20	latmcl 18619	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
44	23, 39, 42, 43	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
45	24, 11	latjcl 18618	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑃) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))
46	23, 37, 44, 45	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))
47	24, 5, 20	latlem12 18645	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝐹‘𝑄) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))) → (((𝐹‘𝑄) ≤ (𝑄 ∨ (𝑅‘𝐹)) ∧ (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))))
48	23, 28, 32, 46, 47	syl13anc 1399	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (((𝐹‘𝑄) ≤ (𝑄 ∨ (𝑅‘𝐹)) ∧ (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))))
49	18, 22, 48	mpbi2and 725	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))
50		hlatl 40510	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
51	1, 50	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ AtLat)
52		simp3r 1221	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ≠ 𝑃)
53	5, 6, 7, 8, 15	trlat 41319	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
54	3, 19, 4, 52, 53	syl112anc 1401	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
55	5, 7, 8, 15	trlle 41334	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
56	3, 4, 55	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ≤ 𝑊)
57		simp23r 1314	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ 𝑄 ≤ 𝑊)
58		nbrne2 5124	. . . . . . 7 ⊢ (((𝑅‘𝐹) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → (𝑅‘𝐹) ≠ 𝑄)
59	58	necomd 3010	. . . . . 6 ⊢ (((𝑅‘𝐹) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → 𝑄 ≠ (𝑅‘𝐹))
60	56, 57, 59	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ≠ (𝑅‘𝐹))
61		eqid 2760	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
62	11, 6, 61	llni2 40662	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅‘𝐹)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾))
63	1, 2, 54, 60, 62	syl31anc 1400	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾))
64	5, 6, 7, 8	ltrnat 41290	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
65	3, 4, 33, 64	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ∈ 𝐴)
66	5, 11, 6	hlatlej1 40525	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
67	1, 33, 65, 66	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
68		simp3l 1220	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)))
69		nbrne2 5124	. . . . . . 7 ⊢ ((𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → 𝑃 ≠ 𝑄)
70	67, 68, 69	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ≠ 𝑄)
71	5, 11, 20, 6, 7	lhpat 41193	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴)
72	3, 19, 2, 70, 71	syl112anc 1401	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴)
73	24, 5, 20	latmle2 18644	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
74	23, 39, 42, 73	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
75	5, 6, 7, 8	ltrnel 41289	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
76	75	simprd 501	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝐹‘𝑃) ≤ 𝑊)
77	3, 4, 19, 76	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ (𝐹‘𝑃) ≤ 𝑊)
78		nbrne2 5124	. . . . . . 7 ⊢ ((((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≠ (𝐹‘𝑃))
79	78	necomd 3010	. . . . . 6 ⊢ ((((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊))
80	74, 77, 79	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊))
81	11, 6, 61	llni2 40662	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) ∧ (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LLines‘𝐾))
82	1, 65, 72, 80, 81	syl31anc 1400	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LLines‘𝐾))
83	5, 11, 20, 6, 7, 8, 15	cdlemc4 41344	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
84	83	3adant3r 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
85	24, 20	latmcl 18619	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ (Base‘𝐾))
86	23, 32, 46, 85	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ (Base‘𝐾))
87		eqid 2760	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
88	24, 5, 87, 6	atlen0 40460	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ (Base‘𝐾) ∧ (𝐹‘𝑄) ∈ 𝐴) ∧ (𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ≠ (0.‘𝐾))
89	51, 86, 10, 49, 88	syl31anc 1400	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ≠ (0.‘𝐾))
90	20, 87, 6, 61	2llnmat 40674	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LLines‘𝐾)) ∧ ((𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∧ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ≠ (0.‘𝐾))) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ 𝐴)
91	1, 63, 82, 84, 89, 90	syl32anc 1405	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ 𝐴)
92	5, 6	atcmp 40461	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ (𝐹‘𝑄) ∈ 𝐴 ∧ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ∈ 𝐴) → ((𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))))
93	51, 10, 91, 92	syl3anc 1398	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝐹‘𝑄) ≤ ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) ↔ (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))))
94	49, 93	mpbid 235	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  lecple 17440  joincjn 18490  meetcmee 18491  0.cp0 18600  Latclat 18610  Atomscatm 40413  AtLatcal 40414  HLchlt 40500  LLinesclln 40641  LHypclh 41134  LTrncltrn 41251  trLctrl 41308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8852  df-proset 18473  df-poset 18492  df-plt 18507  df-lub 18523  df-glb 18524  df-join 18525  df-meet 18526  df-p0 18602  df-p1 18603  df-lat 18611  df-clat 18678  df-oposet 40326  df-ol 40328  df-oml 40329  df-covers 40416  df-ats 40417  df-atl 40448  df-cvlat 40472  df-hlat 40501  df-llines 40648  df-psubsp 40653  df-pmap 40654  df-padd 40946  df-lhyp 41138  df-laut 41139  df-ldil 41254  df-ltrn 41255  df-trl 41309

This theorem is used by:  cdlemc  41347