Theorem cdlemg18b 40658

Description: Lemma for cdlemg18c 40659. TODO: fix comment. (Contributed by NM, 15-May-2013.)

Hypotheses

Ref	Expression
cdlemg12.l	⊢ ≤ = (le‘𝐾)
cdlemg12.j	⊢ ∨ = (join‘𝐾)
cdlemg12.m	⊢ ∧ = (meet‘𝐾)
cdlemg12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg12.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg18b.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)

Assertion

Ref	Expression
cdlemg18b	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))

Proof of Theorem cdlemg18b

Step	Hyp	Ref	Expression
1		simp33 1212	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))
2		simp3r 1203	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))
3		simp1l 1198	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝐾 ∈ HL)
4		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑊 ∈ 𝐻)
5		simp21 1207	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
6		simp22l 1293	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑄 ∈ 𝐴)
7		simp3l1 1279	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑃 ≠ 𝑄)
8		cdlemg12.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
9		cdlemg12.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
10		cdlemg12.m	. . . . . . . . . . . . 13 ⊢ ∧ = (meet‘𝐾)
11		cdlemg12.a	. . . . . . . . . . . . 13 ⊢ 𝐴 = (Atoms‘𝐾)
12		cdlemg12.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (LHyp‘𝐾)
13		cdlemg18b.u	. . . . . . . . . . . . 13 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
14	8, 9, 10, 11, 12, 13	cdleme0a 40190	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
15	3, 4, 5, 6, 7, 14	syl212anc 1382	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑈 ∈ 𝐴)
16		simp1 1136	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simp23 1209	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝐹 ∈ 𝑇)
18		cdlemg12.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
19	8, 11, 12, 18	ltrnat 40119	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
20	16, 17, 6, 19	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝐹‘𝑄) ∈ 𝐴)
21	8, 9, 11	hlatlej1 39354	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → 𝑈 ≤ (𝑈 ∨ (𝐹‘𝑄)))
22	3, 15, 20, 21	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑈 ≤ (𝑈 ∨ (𝐹‘𝑄)))
23	3	hllatd 39343	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝐾 ∈ Lat)
24		simp21l 1291	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑃 ∈ 𝐴)
25		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
26	25, 11	atbase 39268	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
27	24, 26	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑃 ∈ (Base‘𝐾))
28	25, 11	atbase 39268	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
29	15, 28	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → 𝑈 ∈ (Base‘𝐾))
30	25, 9, 11	hlatjcl 39346	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → (𝑈 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾))
31	3, 15, 20, 30	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑈 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾))
32	25, 8, 9	latjle12 18356	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑈 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)) ∧ 𝑈 ≤ (𝑈 ∨ (𝐹‘𝑄))) ↔ (𝑃 ∨ 𝑈) ≤ (𝑈 ∨ (𝐹‘𝑄))))
33	23, 27, 29, 31, 32	syl13anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → ((𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)) ∧ 𝑈 ≤ (𝑈 ∨ (𝐹‘𝑄))) ↔ (𝑃 ∨ 𝑈) ≤ (𝑈 ∨ (𝐹‘𝑄))))
34	2, 22, 33	mpbi2and 712	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑃 ∨ 𝑈) ≤ (𝑈 ∨ (𝐹‘𝑄)))
35	8, 9, 10, 11, 12, 13	cdleme0cp 40193	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄))
36	3, 4, 5, 6, 35	syl22anc 838	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄))
37		simp22 1208	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
38	12, 18, 8, 9, 11, 10, 13	cdlemg2kq 40581	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ 𝑈))
39	16, 5, 37, 17, 38	syl121anc 1377	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ 𝑈))
40	9, 11	hlatjcom 39347	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑄) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝐹‘𝑄) ∨ 𝑈) = (𝑈 ∨ (𝐹‘𝑄)))
41	3, 20, 15, 40	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → ((𝐹‘𝑄) ∨ 𝑈) = (𝑈 ∨ (𝐹‘𝑄)))
42	39, 41	eqtr2d 2765	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑈 ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))
43	34, 36, 42	3brtr3d 5123	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑃 ∨ 𝑄) ≤ ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))
44	8, 11, 12, 18	ltrnat 40119	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
45	16, 17, 24, 44	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝐹‘𝑃) ∈ 𝐴)
46	8, 9, 11	ps-1 39456	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ↔ (𝑃 ∨ 𝑄) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄))))
47	3, 24, 6, 7, 45, 20, 46	syl132anc 1390	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → ((𝑃 ∨ 𝑄) ≤ ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ↔ (𝑃 ∨ 𝑄) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄))))
48	43, 47	mpbid 232	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → (𝑃 ∨ 𝑄) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)))
49	9, 11	hlatjcom 39347	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ (𝐹‘𝑃)))
50	3, 45, 20, 49	syl3anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑄) ∨ (𝐹‘𝑃)))
51	48, 50	eqtr2d 2765	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑄))
52	51	3exp 1119	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → (((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) ∧ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄))) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑄))))
53	52	exp4a 431	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → ((𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄)) → (𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑄)))))
54	53	3imp 1110	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → (𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)) → ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑄)))
55	54	necon3ad 2938	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → (((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄) → ¬ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄))))
56	1, 55	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  Latclat 18337  Atomscatm 39242  HLchlt 39329  LHypclh 39963  LTrncltrn 40080  trLctrl 40137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-riotaBAD 38932

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-undef 8206  df-map 8755  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39155  df-ol 39157  df-oml 39158  df-covers 39245  df-ats 39246  df-atl 39277  df-cvlat 39301  df-hlat 39330  df-llines 39477  df-lplanes 39478  df-lvols 39479  df-lines 39480  df-psubsp 39482  df-pmap 39483  df-padd 39775  df-lhyp 39967  df-laut 39968  df-ldil 40083  df-ltrn 40084  df-trl 40138

This theorem is referenced by:  cdlemg18c  40659