Theorem cdlemg17 41734

Description: Part of Lemma G of [Crawley] p. 117, lines 7 and 8. We show an argument whose value at 𝐺 equals itself. TODO: fix comment. (Contributed by NM, 12-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)

Assertion

Ref	Expression
cdlemg17	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))) = ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))))

Distinct variable groups:   𝐴,𝑟   𝐺,𝑟   ∨ ,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑊,𝑟   𝐹,𝑟

Allowed substitution hints:   𝑅(𝑟)   𝑇(𝑟)   𝐻(𝑟)   𝐾(𝑟)   ∧ (𝑟)

Proof of Theorem cdlemg17

Step	Hyp	Ref	Expression
1		simp11 1222	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp22 1226	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐺 ∈ 𝑇)
3		simp12l 1305	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ∈ 𝐴)
4		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		cdlemg12.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 40346	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
7	3, 6	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ∈ (Base‘𝐾))
8		simp21 1225	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐹 ∈ 𝑇)
9		cdlemg12.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10		cdlemg12.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemg12.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12	9, 5, 10, 11	ltrncoat 41201	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴)
13	1, 8, 2, 3, 12	syl121anc 1402	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴)
14	4, 5	atbase 40346	. . . . . 6 ⊢ ((𝐹‘(𝐺‘𝑃)) ∈ 𝐴 → (𝐹‘(𝐺‘𝑃)) ∈ (Base‘𝐾))
15	13, 14	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑃)) ∈ (Base‘𝐾))
16		cdlemg12.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
17	4, 16, 10, 11	ltrnj 41189	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘(𝐺‘𝑃)) ∈ (Base‘𝐾))) → (𝐺‘(𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) = ((𝐺‘𝑃) ∨ (𝐺‘(𝐹‘(𝐺‘𝑃)))))
18	1, 2, 7, 15, 17	syl112anc 1401	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) = ((𝐺‘𝑃) ∨ (𝐺‘(𝐹‘(𝐺‘𝑃)))))
19		simp1 1154	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
20		simp23 1227	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ≠ 𝑄)
21		simp3 1156	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟))))
22		cdlemg12.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
23		cdlemg12b.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
24	9, 16, 22, 5, 10, 11, 23	cdlemg17b 41719	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘𝑃) = 𝑄)
25	19, 2, 20, 21, 24	syl121anc 1402	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘𝑃) = 𝑄)
26	25	fveq2d 6889	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑃)) = (𝐹‘𝑄))
27	26	fveq2d 6889	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘(𝐺‘𝑃))) = (𝐺‘(𝐹‘𝑄)))
28	9, 16, 22, 5, 10, 11, 23	cdlemg17jq 41733	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘𝑄)) = (𝐹‘(𝐺‘𝑄)))
29	27, 28	eqtrd 2796	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘(𝐺‘𝑃))) = (𝐹‘(𝐺‘𝑄)))
30	25, 29	oveq12d 7438	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝐺‘𝑃) ∨ (𝐺‘(𝐹‘(𝐺‘𝑃)))) = (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))
31	18, 30	eqtrd 2796	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) = (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))
32		simp13l 1307	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑄 ∈ 𝐴)
33	4, 5	atbase 40346	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
34	32, 33	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑄 ∈ (Base‘𝐾))
35	9, 5, 10, 11	ltrncoat 41201	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴)
36	1, 8, 2, 32, 35	syl121anc 1402	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴)
37	4, 5	atbase 40346	. . . . . 6 ⊢ ((𝐹‘(𝐺‘𝑄)) ∈ 𝐴 → (𝐹‘(𝐺‘𝑄)) ∈ (Base‘𝐾))
38	36, 37	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑄)) ∈ (Base‘𝐾))
39	4, 16, 10, 11	ltrnj 41189	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐹‘(𝐺‘𝑄)) ∈ (Base‘𝐾))) → (𝐺‘(𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) = ((𝐺‘𝑄) ∨ (𝐺‘(𝐹‘(𝐺‘𝑄)))))
40	1, 2, 34, 38, 39	syl112anc 1401	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) = ((𝐺‘𝑄) ∨ (𝐺‘(𝐹‘(𝐺‘𝑄)))))
41	9, 16, 22, 5, 10, 11, 23	cdlemg17bq 41730	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘𝑄) = 𝑃)
42	41	fveq2d 6889	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐹‘(𝐺‘𝑄)) = (𝐹‘𝑃))
43	42	fveq2d 6889	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘(𝐺‘𝑄))) = (𝐺‘(𝐹‘𝑃)))
44	9, 16, 22, 5, 10, 11, 23	cdlemg17j 41728	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘𝑃)) = (𝐹‘(𝐺‘𝑃)))
45	43, 44	eqtrd 2796	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘(𝐺‘𝑄))) = (𝐹‘(𝐺‘𝑃)))
46	41, 45	oveq12d 7438	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝐺‘𝑄) ∨ (𝐺‘(𝐹‘(𝐺‘𝑄)))) = (𝑃 ∨ (𝐹‘(𝐺‘𝑃))))
47	40, 46	eqtrd 2796	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) = (𝑃 ∨ (𝐹‘(𝐺‘𝑃))))
48	31, 47	oveq12d 7438	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝐺‘(𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) ∧ (𝐺‘(𝑄 ∨ (𝐹‘(𝐺‘𝑄))))) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ (𝑃 ∨ (𝐹‘(𝐺‘𝑃)))))
49		simp11l 1303	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐾 ∈ HL)
50	4, 16, 5	hlatjcl 40424	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) → (𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾))
51	49, 3, 13, 50	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾))
52	4, 16, 5	hlatjcl 40424	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑄)) ∈ 𝐴) → (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾))
53	49, 32, 36, 52	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾))
54	4, 22, 10, 11	ltrnm 41188	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾))) → (𝐺‘((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))) = ((𝐺‘(𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) ∧ (𝐺‘(𝑄 ∨ (𝐹‘(𝐺‘𝑄))))))
55	1, 2, 51, 53, 54	syl112anc 1401	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))) = ((𝐺‘(𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) ∧ (𝐺‘(𝑄 ∨ (𝐹‘(𝐺‘𝑄))))))
56	49	hllatd 40421	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐾 ∈ Lat)
57	4, 22	latmcom 18637	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ (𝑃 ∨ (𝐹‘(𝐺‘𝑃)))))
58	56, 51, 53, 57	syl3anc 1398	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ (𝑃 ∨ (𝐹‘(𝐺‘𝑃)))))
59	48, 55, 58	3eqtr4d 2806	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))) = ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  lecple 17435  joincjn 18485  meetcmee 18486  Latclat 18605  Atomscatm 40320  HLchlt 40407  LHypclh 41041  LTrncltrn 41158  trLctrl 41215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-riotaBAD 40010

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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-undef 8290  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40233  df-ol 40235  df-oml 40236  df-covers 40323  df-ats 40324  df-atl 40355  df-cvlat 40379  df-hlat 40408  df-llines 40555  df-lplanes 40556  df-lvols 40557  df-lines 40558  df-psubsp 40560  df-pmap 40561  df-padd 40853  df-lhyp 41045  df-laut 41046  df-ldil 41161  df-ltrn 41162  df-trl 41216

This theorem is used by:  cdlemg18  41739