Theorem cdlemc5 38136

Description: Lemma for cdlemc 38138. (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 1195	. . . . 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 1292	. . . . 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 1134	. . . . . 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 1204	. . . . . 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 38081	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
10	3, 4, 2, 9	syl3anc 1369	. . . . 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 37317	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → (𝐹‘𝑄) ≤ (𝑄 ∨ (𝐹‘𝑄)))
13	1, 2, 10, 12	syl3anc 1369	. . . 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 1206	. . . . 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 38107	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (𝐹‘𝑄)))
17	3, 4, 14, 16	syl3anc 1369	. . . 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 5098	. . 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 1205	. . . 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 38133	. . . 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 1372	. . 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 37305	. . . 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 2738	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 6	atbase 37230	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
26	2, 25	syl 17	. . . . 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 38066	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐹‘𝑄) ∈ (Base‘𝐾))
28	3, 4, 26, 27	syl3anc 1369	. . . 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 38105	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	3, 4, 29	syl2anc 583	. . . . 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 18072	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (Base‘𝐾))
32	23, 26, 30, 31	syl3anc 1369	. . . 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 1290	. . . . . . 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 37230	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
35	33, 34	syl 17	. . . . . 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 38066	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
37	3, 4, 35, 36	syl3anc 1369	. . . . 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 37308	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
39	1, 33, 2, 38	syl3anc 1369	. . . . . 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 1196	. . . . . . 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 37939	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
42	40, 41	syl 17	. . . . . 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 18073	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
44	23, 39, 42, 43	syl3anc 1369	. . . . 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 18072	. . . . 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 1369	. . . 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 18099	. . . 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 1370	. . 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 708	. 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 37301	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
51	1, 50	syl 17	. . 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 1200	. . . . . 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 38110	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
54	3, 19, 4, 52, 53	syl112anc 1372	. . . . 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 38125	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
56	3, 4, 55	syl2anc 583	. . . . . 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 1293	. . . . . 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 5090	. . . . . . 7 ⊢ (((𝑅‘𝐹) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → (𝑅‘𝐹) ≠ 𝑄)
59	58	necomd 2998	. . . . . 6 ⊢ (((𝑅‘𝐹) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → 𝑄 ≠ (𝑅‘𝐹))
60	56, 57, 59	syl2anc 583	. . . . 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 2738	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
62	11, 6, 61	llni2 37453	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅‘𝐹)) → (𝑄 ∨ (𝑅‘𝐹)) ∈ (LLines‘𝐾))
63	1, 2, 54, 60, 62	syl31anc 1371	. . . 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 38081	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
65	3, 4, 33, 64	syl3anc 1369	. . . . 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 37316	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
67	1, 33, 65, 66	syl3anc 1369	. . . . . . 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 1199	. . . . . . 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 5090	. . . . . . 7 ⊢ ((𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → 𝑃 ≠ 𝑄)
70	67, 68, 69	syl2anc 583	. . . . . 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 37984	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴)
72	3, 19, 2, 70, 71	syl112anc 1372	. . . . 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 18098	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
74	23, 39, 42, 73	syl3anc 1369	. . . . . 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 38080	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
76	75	simprd 495	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝐹‘𝑃) ≤ 𝑊)
77	3, 4, 19, 76	syl3anc 1369	. . . . . 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 5090	. . . . . . 7 ⊢ ((((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≠ (𝐹‘𝑃))
79	78	necomd 2998	. . . . . 6 ⊢ ((((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → (𝐹‘𝑃) ≠ ((𝑃 ∨ 𝑄) ∧ 𝑊))
80	74, 77, 79	syl2anc 583	. . . . 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 37453	. . . . 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 1371	. . . 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 38135	. . . . 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 1179	. . . 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 18073	. . . . . 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 1369	. . . . 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 2738	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
88	24, 5, 87, 6	atlen0 37251	. . . . 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 1371	. . . 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 37465	. . . 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 1376	. . 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 37252	. . 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 1369	. 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 231	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))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  meetcmee 17945  0.cp0 18056  Latclat 18064  Atomscatm 37204  AtLatcal 37205  HLchlt 37291  LLinesclln 37432  LHypclh 37925  LTrncltrn 38042  trLctrl 38099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100

This theorem is referenced by:  cdlemc  38138