Theorem cdlemm10N 38775

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 7131	. . . . 5 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) ∈ V
2		cdlemm10.g	. . . . 5 ⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞))
3	1, 2	fnmpti 6480	. . . 4 ⊢ 𝐺 Fn 𝐶
4		fvelrnb 6730	. . . 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 2750	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑠 → ((𝑓‘𝑃) = 𝑞 ↔ (𝑓‘𝑃) = 𝑠))
7	6	riotabidv 7129	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑠 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
8		riotaex 7131	. . . . . . . . . . 11 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) ∈ V
9	7, 2, 8	fvmpt 6775	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
10		cdlemm10.f	. . . . . . . . . 10 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)
11	9, 10	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = 𝐹)
12	11	adantl 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐺‘𝑠) = 𝐹)
13	12	eqeq1d 2740	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → ((𝐺‘𝑠) = 𝑔 ↔ 𝐹 = 𝑔))
14	13	rexbidva 3206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (∃𝑠 ∈ 𝐶 (𝐺‘𝑠) = 𝑔 ↔ ∃𝑠 ∈ 𝐶 𝐹 = 𝑔))
15		simpl1 1192	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		simprl 771	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝑔 ∈ 𝑇)
17		simpl2l 1227	. . . . . . . . . . 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 37797	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑔‘𝑃) ∈ 𝐴)
23	15, 16, 17, 22	syl3anc 1372	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ 𝐴)
24		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
25		simpl1l 1225	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ HL)
26	25	hllatd 37021	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ Lat)
27	24, 19	atbase 36946	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
28	17, 27	syl 17	. . . . . . . . . . . . 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 37782	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑔‘𝑃) ∈ (Base‘𝐾))
30	15, 16, 28, 29	syl3anc 1372	. . . . . . . . . . . 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 17777	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑔‘𝑃)) ∈ (Base‘𝐾))
33	26, 28, 30, 32	syl3anc 1372	. . . . . . . . . . . 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 1229	. . . . . . . . . . . . 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 37024	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
36	25, 17, 34, 35	syl3anc 1372	. . . . . . . . . . . 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 17787	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑔‘𝑃) ≤ (𝑃 ∨ (𝑔‘𝑃)))
38	26, 28, 30, 37	syl3anc 1372	. . . . . . . . . . . 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 1193	. . . . . . . . . . . . . 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 37823	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑔‘𝑃)))
42	15, 16, 39, 41	syl3anc 1372	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 37821	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (𝑅‘𝑔) ∈ (Base‘𝐾))
45	15, 16, 44	syl2anc 587	. . . . . . . . . . . . . . 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 36946	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
47	34, 46	syl 17	. . . . . . . . . . . . . . 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 17792	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 15	. . . . . . . . . . . . 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 5054	. . . . . . . . . . . 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 17784	. . . . . . . . . . 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 37796	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑔‘𝑃) ∈ 𝐴 ∧ ¬ (𝑔‘𝑃) ≤ 𝑊))
54	53	simprd 499	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝑔‘𝑃) ≤ 𝑊)
55	15, 16, 39, 54	syl3anc 1372	. . . . . . . . . . 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 515	. . . . . . . . . 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 5033	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉)))
58		breq1 5033	. . . . . . . . . . . . 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 634	. . . . . . . . . . 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 3591	. . . . . . . . . 10 ⊢ ((𝑔‘𝑃) ∈ 𝐶 ↔ ((𝑔‘𝑃) ∈ 𝐴 ∧ ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊)))
63	23, 56, 62	sylanbrc 586	. . . . . . . . 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 38242	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑔 ∈ 𝑇) → 𝑔 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
65	15, 39, 16, 64	syl3anc 1372	. . . . . . . . . 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 2744	. . . . . . . . 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 2750	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑔‘𝑃) → ((𝑓‘𝑃) = 𝑠 ↔ (𝑓‘𝑃) = (𝑔‘𝑃)))
68	67	riotabidv 7129	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑔‘𝑃) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
69	10, 68	syl5eq 2785	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑔‘𝑃) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
70	69	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑠 = (𝑔‘𝑃) → (𝐹 = 𝑔 ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔))
71	70	rspcev 3526	. . . . . . . . 9 ⊢ (((𝑔‘𝑃) ∈ 𝐶 ∧ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)
72	63, 66, 71	syl2anc 587	. . . . . . . 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 416	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔))
74		breq1 5033	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ 𝑠 ≤ (𝑃 ∨ 𝑉)))
75		breq1 5033	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ 𝑊 ↔ 𝑠 ≤ 𝑊))
76	75	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑠 ≤ 𝑊))
77	74, 76	anbi12d 634	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → ((𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊) ↔ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)))
78	77, 61	elrab2 3591	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐶 ↔ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)))
79		simpl1 1192	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
81		simpl2r 1228	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → ¬ 𝑃 ≤ 𝑊)
82		simprl 771	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ∈ 𝐴)
83		simprrr 782	. . . . . . . . . . . . 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 38236	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
85	18, 19, 20, 21, 10	ltrniotaval 38238	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑠)
86	84, 85	jca 515	. . . . . . . . . . . . 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 1380	. . . . . . . . . . . 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 1202	. . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (meet‘𝐾) = (meet‘𝐾)
92	18, 31, 91, 19, 20, 21, 40	trlval2 37820	. . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . 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 7185	. . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 1225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝐾 ∈ HL)
99		simpl3l 1229	. . . . . . . . . . . . . . . . . . 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 37032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑉))
101	98, 80, 99, 100	syl3anc 1372	. . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . 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 37021	. . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . 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 36946	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
106	105	ad2antrl 728	. . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . 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 17788	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . 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 37024	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → (𝑃 ∨ 𝑠) ∈ (Base‘𝐾))
112	98, 80, 82, 111	syl3anc 1372	. . . . . . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . . . . . . 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 37655	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
115	113, 114	syl 17	. . . . . . . . . . . . . . . . . 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 17807	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . 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 37701	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊) = 𝑉)
120	119	adantr 484	. . . . . . . . . . . . . . . 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 5056	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 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 5052	. . . . . . . . . . . . 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 515	. . . . . . . . . . . 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 1463	. . . . . . . . . . 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 597	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉))
127	126	ex 416	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑠 ∈ 𝐶 → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉)))
128		eleq1 2820	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑔 → (𝐹 ∈ 𝑇 ↔ 𝑔 ∈ 𝑇))
129		fveq2 6674	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑔 → (𝑅‘𝐹) = (𝑅‘𝑔))
130	129	breq1d 5040	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑔 → ((𝑅‘𝐹) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
131	128, 130	anbi12d 634	. . . . . . . . . 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 35	. . . . . . . 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 3193	. . . . . . 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 6674	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑅‘𝑓) = (𝑅‘𝑔))
138	137	breq1d 5040	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑅‘𝑓) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
139	138	elrab 3588	. . . . 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 1198	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝐾 ∈ HL)
142		simp1r 1199	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
143		simp3l 1202	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ 𝐴)
144	143, 46	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → 𝑉 ∈ (Base‘𝐾))
145		simp3r 1203	. . . . . 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 38689	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉})
148	141, 142, 144, 145, 147	syl22anc 838	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉})
149	148	eleq2d 2818	. . . 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	syl5bb 286	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑔 ∈ ran 𝐺 ↔ 𝑔 ∈ (𝐼‘𝑉)))
152	151	eqrdv 2736	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∃wrex 3054  {crab 3057   class class class wbr 5030   ↦ cmpt 5110  ran crn 5526   Fn wfn 6334  ‘cfv 6339  ℩crio 7126  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  meetcmee 17671  Latclat 17771  Atomscatm 36920  HLchlt 37007  LHypclh 37641  LTrncltrn 37758  trLctrl 37815  DIsoAcdia 38685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-riotaBAD 36610

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-undef 7968  df-map 8439  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36833  df-ol 36835  df-oml 36836  df-covers 36923  df-ats 36924  df-atl 36955  df-cvlat 36979  df-hlat 37008  df-llines 37155  df-lplanes 37156  df-lvols 37157  df-lines 37158  df-psubsp 37160  df-pmap 37161  df-padd 37453  df-lhyp 37645  df-laut 37646  df-ldil 37761  df-ltrn 37762  df-trl 37816  df-disoa 38686

This theorem is referenced by: (None)