Theorem cdlemm10N 42175

Description: The image of the map 𝐺 is the entire one-dimensional subspace (𝐼‘𝑉). Remark after Lemma M of [Crawley] p. 121 line 23. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemm10.l	⊢ ≤ = (le‘𝐾)
cdlemm10.j	⊢ ∨ = (join‘𝐾)
cdlemm10.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemm10.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemm10.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemm10.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemm10.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
cdlemm10.c	⊢ 𝐶 = {𝑟 ∈ 𝐴 ∣ (𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊)}
cdlemm10.f	⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)
cdlemm10.g	⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞))

Assertion

Ref	Expression
cdlemm10N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉))

Distinct variable groups:   𝑓,𝑟,𝑠, ≤   ∨ ,𝑟   𝐴,𝑓,𝑟,𝑠   𝑠,𝑞,𝐶   𝐺,𝑠   𝑓,𝐻,𝑠   𝑓,𝐾,𝑠   𝑓,𝑞,𝑃,𝑟,𝑠   𝑅,𝑓,𝑠   𝑇,𝑓,𝑞,𝑠   𝑓,𝑉,𝑟,𝑠   𝑓,𝑊,𝑟,𝑠

Allowed substitution hints:   𝐴(𝑞)   𝐶(𝑓, 𝑟)   𝑅(𝑟, 𝑞)   𝑇(𝑟)   𝐹(𝑓, 𝑠, 𝑟, 𝑞)   𝐺(𝑓, 𝑟, 𝑞)   𝐻(𝑟, 𝑞)   𝐼(𝑓, 𝑠, 𝑟, 𝑞)   ∨ (𝑓, 𝑠, 𝑞)   𝐾(𝑟, 𝑞)   ≤ (𝑞)   𝑉(𝑞)   𝑊(𝑞)

Proof of Theorem cdlemm10N

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 7381	. . . . 5 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) ∈ V
2		cdlemm10.g	. . . . 5 ⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞))
3	1, 2	fnmpti 6682	. . . 4 ⊢ 𝐺 Fn 𝐶
4		fvelrnb 6945	. . . 4 ⊢ (𝐺 Fn 𝐶 → (𝑔 ∈ ran 𝐺 ↔ ∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔))
5	3, 4	ax-mp 5	. . 3 ⊢ (𝑔 ∈ ran 𝐺 ↔ ∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔)
6		eqeq2 2773	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑠 → ((𝑓‘𝑃) = 𝑞 ↔ (𝑓‘𝑃) = 𝑠))
7	6	riotabidv 7379	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑠 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
8		riotaex 7381	. . . . . . . . . . 11 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) ∈ V
9	7, 2, 8	fvmpt 6993	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
10		cdlemm10.f	. . . . . . . . . 10 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)
11	9, 10	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = 𝐹)
12	11	adantl 487	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐺‘𝑠) = 𝐹)
13	12	eqeq1d 2763	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → ((𝐺‘𝑠) = 𝑔 ↔ 𝐹 = 𝑔))
14	13	rexbidva 3185	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ ∃𝑠 ∈ 𝐶 𝐹 = 𝑔))
15		simpl1 1210	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑔 ∈ 𝑇)
17		simpl2l 1245	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑃 ∈ 𝐴)
18		cdlemm10.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
19		cdlemm10.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
20		cdlemm10.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (LHyp‘𝐾)
21		cdlemm10.t	. . . . . . . . . . . 12 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
22	18, 19, 20, 21	ltrnat 41197	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑔‘𝑃) ∈ 𝐴)
23	15, 16, 17, 22	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ 𝐴)
24		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
25		simpl1l 1243	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ HL)
26	25	hllatd 40421	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ Lat)
27	24, 19	atbase 40346	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
28	17, 27	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑃 ∈ (Base‘𝐾))
29	24, 20, 21	ltrncl 41182	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑔‘𝑃) ∈ (Base‘𝐾))
30	15, 16, 28, 29	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ (Base‘𝐾))
31		cdlemm10.j	. . . . . . . . . . . . . 14 ⊢ ∨ = (join‘𝐾)
32	24, 31	latjcl 18613	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑔‘𝑃)) ∈ (Base‘𝐾))
33	26, 28, 30, 32	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑔‘𝑃)) ∈ (Base‘𝐾))
34		simpl3l 1247	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑉 ∈ 𝐴)
35	24, 31, 19	hlatjcl 40424	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
36	25, 17, 34, 35	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
37	24, 18, 31	latlej2 18623	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑔‘𝑃) ≤ (𝑃 ∨ (𝑔‘𝑃)))
38	26, 28, 30, 37	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ≤ (𝑃 ∨ (𝑔‘𝑃)))
39		simpl2 1211	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
40		cdlemm10.r	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
41	18, 31, 19, 20, 21, 40	trljat1 41223	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑔‘𝑃)))
42	15, 16, 39, 41	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑔‘𝑃)))
43		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑅‘𝑔) ≤ 𝑉)
44	24, 20, 21, 40	trlcl 41221	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (𝑅‘𝑔) ∈ (Base‘𝐾))
45	15, 16, 44	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑅‘𝑔) ∈ (Base‘𝐾))
46	24, 19	atbase 40346	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
47	34, 46	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑉 ∈ (Base‘𝐾))
48	24, 18, 31	latjlej2 18628	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝑔) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → ((𝑅‘𝑔) ≤ 𝑉 → (𝑃 ∨ (𝑅‘𝑔)) ≤ (𝑃 ∨ 𝑉)))
49	26, 45, 47, 28, 48	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ((𝑅‘𝑔) ≤ 𝑉 → (𝑃 ∨ (𝑅‘𝑔)) ≤ (𝑃 ∨ 𝑉)))
50	43, 49	mpd 16	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑅‘𝑔)) ≤ (𝑃 ∨ 𝑉))
51	42, 50	eqbrtrrd 5129	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑃 ∨ (𝑔‘𝑃)) ≤ (𝑃 ∨ 𝑉))
52	24, 18, 26, 30, 33, 36, 38, 51	lattrd 18620	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉))
53	18, 19, 20, 21	ltrnel 41196	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑔‘𝑃) ∈ 𝐴 ∧ ¬ (𝑔‘𝑃) ≤ 𝑊))
54	53	simprd 501	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝑔‘𝑃) ≤ 𝑊)
55	15, 16, 39, 54	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ¬ (𝑔‘𝑃) ≤ 𝑊)
56	52, 55	jca 521	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊))
57		breq1 5106	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉)))
58		breq1 5106	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ 𝑊 ↔ (𝑔‘𝑃) ≤ 𝑊))
59	58	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑔‘𝑃) → (¬ 𝑟 ≤ 𝑊 ↔ ¬ (𝑔‘𝑃) ≤ 𝑊))
60	57, 59	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑔‘𝑃) → ((𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊) ↔ ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊)))
61		cdlemm10.c	. . . . . . . . . . 11 ⊢ 𝐶 = {𝑟 ∈ 𝐴 ∣ (𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊)}
62	60, 61	elrab2 3649	. . . . . . . . . 10 ⊢ ((𝑔‘𝑃) ∈ 𝐶 ↔ ((𝑔‘𝑃) ∈ 𝐴 ∧ ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊)))
63	23, 56, 62	sylanbrc 595	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ 𝐶)
64	18, 19, 20, 21	cdlemeiota 41642	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑔 ∈ 𝑇) → 𝑔 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
65	15, 39, 16, 64	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑔 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
66	65	eqcomd 2767	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔)
67		eqeq2 2773	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑔‘𝑃) → ((𝑓‘𝑃) = 𝑠 ↔ (𝑓‘𝑃) = (𝑔‘𝑃)))
68	67	riotabidv 7379	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑔‘𝑃) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
69	10, 68	eqtrid 2808	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑔‘𝑃) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
70	69	eqeq1d 2763	. . . . . . . . . 10 ⊢ (𝑠 = (𝑔‘𝑃) → (𝐹 = 𝑔 ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔))
71	70	rspcev 3577	. . . . . . . . 9 ⊢ (((𝑔‘𝑃) ∈ 𝐶 ∧ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)
72	63, 66, 71	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)
73	72	ex 418	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔))
74		breq1 5106	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ 𝑠 ≤ (𝑃 ∨ 𝑉)))
75		breq1 5106	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ 𝑊 ↔ 𝑠 ≤ 𝑊))
76	75	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑠 ≤ 𝑊))
77	74, 76	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → ((𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊) ↔ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)))
78	77, 61	elrab2 3649	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐶 ↔ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)))
79		simpl1 1210	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
80		simpl2l 1245	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
81		simpl2r 1246	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ¬ 𝑃 ≤ 𝑊)
82		simprl 783	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ∈ 𝐴)
83		simprrr 794	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ¬ 𝑠 ≤ 𝑊)
84	18, 19, 20, 21, 10	ltrniotacl 41636	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
85	18, 19, 20, 21, 10	ltrniotaval 41638	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑠)
86	84, 85	jca 521	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠))
87	79, 80, 81, 82, 83, 86	syl122anc 1406	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠))
88		simp3l 1220	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → 𝐹 ∈ 𝑇)
89		simp11 1222	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
90		simp12 1223	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
91		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (meet‘𝐾) = (meet‘𝐾)
92	18, 31, 91, 19, 20, 21, 40	trlval2 41220	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊))
93	89, 88, 90, 92	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊))
94		simp3r 1221	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝐹‘𝑃) = 𝑠)
95	94	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑃 ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑠))
96	95	oveq1d 7435	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → ((𝑃 ∨ (𝐹‘𝑃))(meet‘𝐾)𝑊) = ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊))
97	93, 96	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑅‘𝐹) = ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊))
98		simpl1l 1243	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝐾 ∈ HL)
99		simpl3l 1247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑉 ∈ 𝐴)
100	18, 31, 19	hlatlej1 40432	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑉))
101	98, 80, 99, 100	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ≤ (𝑃 ∨ 𝑉))
102		simprrl 793	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ≤ (𝑃 ∨ 𝑉))
103	98	hllatd 40421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝐾 ∈ Lat)
104	80, 27	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ∈ (Base‘𝐾))
105	24, 19	atbase 40346	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
106	105	ad2antrl 741	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ∈ (Base‘𝐾))
107	98, 80, 99, 35	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
108	24, 18, 31	latjle12 18624	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑠 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑉) ∧ 𝑠 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉)))
109	103, 104, 106, 107, 108	syl13anc 1399	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ≤ (𝑃 ∨ 𝑉) ∧ 𝑠 ≤ (𝑃 ∨ 𝑉)) ↔ (𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉)))
110	101, 102, 109	mpbi2and 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉))
111	24, 31, 19	hlatjcl 40424	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → (𝑃 ∨ 𝑠) ∈ (Base‘𝐾))
112	98, 80, 82, 111	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝑃 ∨ 𝑠) ∈ (Base‘𝐾))
113		simpl1r 1244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑊 ∈ 𝐻)
114	24, 20	lhpbase 41055	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
115	113, 114	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑊 ∈ (Base‘𝐾))
116	24, 18, 91	latmlem1 18643	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑠) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊)))
117	103, 112, 107, 115, 116	syl13anc 1399	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑠) ≤ (𝑃 ∨ 𝑉) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊)))
118	110, 117	mpd 16	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊))
119	18, 31, 91, 19, 20	lhpat4N 41101	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊) = 𝑉)
120	119	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊) = 𝑉)
121	118, 120	breqtrd 5131	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ 𝑉)
122	121	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → ((𝑃 ∨ 𝑠)(meet‘𝐾)𝑊) ≤ 𝑉)
123	97, 122	eqbrtrd 5127	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝑅‘𝐹) ≤ 𝑉)
124	88, 123	jca 521	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑠)) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉))
125	87, 124	mpd3an3 1491	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉))
126	78, 125	sylan2b 606	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉))
127	126	ex 418	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑠 ∈ 𝐶 → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉)))
128		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑔 → (𝐹 ∈ 𝑇 ↔ 𝑔 ∈ 𝑇))
129		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑔 → (𝑅‘𝐹) = (𝑅‘𝑔))
130	129	breq1d 5113	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑔 → ((𝑅‘𝐹) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
131	128, 130	anbi12d 644	. . . . . . . . . 10 ⊢ (𝐹 = 𝑔 → ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉) ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)))
132	131	biimpcd 252	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉) → (𝐹 = 𝑔 → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)))
133	127, 132	syl6 36	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑠 ∈ 𝐶 → (𝐹 = 𝑔 → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))))
134	133	rexlimdv 3162	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 𝐹 = 𝑔 → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)))
135	73, 134	impbid 215	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉) ↔ ∃𝑠 ∈ 𝐶 𝐹 = 𝑔))
136	14, 135	bitr4d 285	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)))
137		fveq2 6885	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑅‘𝑓) = (𝑅‘𝑔))
138	137	breq1d 5113	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑅‘𝑓) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
139	138	elrab 3645	. . . . 5 ⊢ (𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉} ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))
140	136, 139	bitr4di 292	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ 𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉}))
141		simp1l 1216	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝐾 ∈ HL)
142		simp1r 1217	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
143		simp3l 1220	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ 𝐴)
144	143, 46	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ (Base‘𝐾))
145		simp3r 1221	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ≤ 𝑊)
146		cdlemm10.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
147	24, 18, 20, 21, 40, 146	diaval 42089	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉})
148	141, 142, 144, 145, 147	syl22anc 852	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉})
149	148	eleq2d 2847	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑔 ∈ (𝐼‘𝑉) ↔ 𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉}))
150	140, 149	bitr4d 285	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ 𝑔 ∈ (𝐼‘𝑉)))
151	5, 150	bitrid 286	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑔 ∈ ran 𝐺 ↔ 𝑔 ∈ (𝐼‘𝑉)))
152	151	eqrdv 2759	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  {crab 3413   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652   Fn wfn 6533  ‘cfv 6538  ℩crio 7376  (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  DIsoAcdia 42085

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  df-disoa 42086

This theorem is used by: (None)