Theorem cdlemm10N 41112

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 7348	. . . . 5 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) ∈ V
2		cdlemm10.g	. . . . 5 ⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞))
3	1, 2	fnmpti 6661	. . . 4 ⊢ 𝐺 Fn 𝐶
4		fvelrnb 6921	. . . 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 2741	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑠 → ((𝑓‘𝑃) = 𝑞 ↔ (𝑓‘𝑃) = 𝑠))
7	6	riotabidv 7346	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑠 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
8		riotaex 7348	. . . . . . . . . . 11 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) ∈ V
9	7, 2, 8	fvmpt 6968	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
10		cdlemm10.f	. . . . . . . . . 10 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)
11	9, 10	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝐶 → (𝐺‘𝑠) = 𝐹)
12	11	adantl 481	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐺‘𝑠) = 𝐹)
13	12	eqeq1d 2731	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → ((𝐺‘𝑠) = 𝑔 ↔ 𝐹 = 𝑔))
14	13	rexbidva 3155	. . . . . 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 770	. . . . . . . . . . 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 40134	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑔‘𝑃) ∈ 𝐴)
23	15, 16, 17, 22	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → (𝑔‘𝑃) ∈ 𝐴)
24		eqid 2729	. . . . . . . . . . . 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 39357	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ Lat)
27	24, 19	atbase 39282	. . . . . . . . . . . . . 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 40119	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑔‘𝑃) ∈ (Base‘𝐾))
30	15, 16, 28, 29	syl3anc 1373	. . . . . . . . . . . 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 18398	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝑔‘𝑃)) ∈ (Base‘𝐾))
33	26, 28, 30, 32	syl3anc 1373	. . . . . . . . . . . 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 39360	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑃 ∨ 𝑉) ∈ (Base‘𝐾))
36	25, 17, 34, 35	syl3anc 1373	. . . . . . . . . . . 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 18408	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑔‘𝑃) ∈ (Base‘𝐾)) → (𝑔‘𝑃) ≤ (𝑃 ∨ (𝑔‘𝑃)))
38	26, 28, 30, 37	syl3anc 1373	. . . . . . . . . . . 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 40160	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑔‘𝑃)))
42	15, 16, 39, 41	syl3anc 1373	. . . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 40158	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (𝑅‘𝑔) ∈ (Base‘𝐾))
45	15, 16, 44	syl2anc 584	. . . . . . . . . . . . . . 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 39282	. . . . . . . . . . . . . . . 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 18413	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 5131	. . . . . . . . . . . 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 18405	. . . . . . . . . . 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 40133	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑔‘𝑃) ∈ 𝐴 ∧ ¬ (𝑔‘𝑃) ≤ 𝑊))
54	53	simprd 495	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ¬ (𝑔‘𝑃) ≤ 𝑊)
55	15, 16, 39, 54	syl3anc 1373	. . . . . . . . . . 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 511	. . . . . . . . . 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 5110	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉)))
58		breq1 5110	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ 𝑊 ↔ (𝑔‘𝑃) ≤ 𝑊))
59	58	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑔‘𝑃) → (¬ 𝑟 ≤ 𝑊 ↔ ¬ (𝑔‘𝑃) ≤ 𝑊))
60	57, 59	anbi12d 632	. . . . . . . . . . 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 3662	. . . . . . . . . 10 ⊢ ((𝑔‘𝑃) ∈ 𝐶 ↔ ((𝑔‘𝑃) ∈ 𝐴 ∧ ((𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉) ∧ ¬ (𝑔‘𝑃) ≤ 𝑊)))
63	23, 56, 62	sylanbrc 583	. . . . . . . . 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 40579	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑔 ∈ 𝑇) → 𝑔 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
65	15, 39, 16, 64	syl3anc 1373	. . . . . . . . . 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 2735	. . . . . . . . 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 2741	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑔‘𝑃) → ((𝑓‘𝑃) = 𝑠 ↔ (𝑓‘𝑃) = (𝑔‘𝑃)))
68	67	riotabidv 7346	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑔‘𝑃) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
69	10, 68	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝑠 = (𝑔‘𝑃) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)))
70	69	eqeq1d 2731	. . . . . . . . . 10 ⊢ (𝑠 = (𝑔‘𝑃) → (𝐹 = 𝑔 ↔ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔))
71	70	rspcev 3588	. . . . . . . . 9 ⊢ (((𝑔‘𝑃) ∈ 𝐶 ∧ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = (𝑔‘𝑃)) = 𝑔) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔)
72	63, 66, 71	syl2anc 584	. . . . . . . 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 412	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉) → ∃𝑠 ∈ 𝐶 𝐹 = 𝑔))
74		breq1 5110	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ 𝑠 ≤ (𝑃 ∨ 𝑉)))
75		breq1 5110	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ 𝑊 ↔ 𝑠 ≤ 𝑊))
76	75	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (¬ 𝑟 ≤ 𝑊 ↔ ¬ 𝑠 ≤ 𝑊))
77	74, 76	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑠 → ((𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊) ↔ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊)))
78	77, 61	elrab2 3662	. . . . . . . . . . 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 770	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐴 ∧ (𝑠 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑠 ≤ 𝑊))) → 𝑠 ∈ 𝐴)
83		simprrr 781	. . . . . . . . . . . . 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 40573	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
85	18, 19, 20, 21, 10	ltrniotaval 40575	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑠)
86	84, 85	jca 511	. . . . . . . . . . . . 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 1381	. . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . 17 ⊢ (meet‘𝐾) = (meet‘𝐾)
92	18, 31, 91, 19, 20, 21, 40	trlval2 40157	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 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 39368	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑉))
101	98, 80, 99, 100	syl3anc 1373	. . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . 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 39357	. . . . . . . . . . . . . . . . . . 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 39282	. . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . 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 18409	. . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . 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 39360	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → (𝑃 ∨ 𝑠) ∈ (Base‘𝐾))
112	98, 80, 82, 111	syl3anc 1373	. . . . . . . . . . . . . . . . . 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 39992	. . . . . . . . . . . . . . . . . . 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 18428	. . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . 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 40038	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ((𝑃 ∨ 𝑉)(meet‘𝐾)𝑊) = 𝑉)
120	119	adantr 480	. . . . . . . . . . . . . . . 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 5133	. . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . 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 5129	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 1464	. . . . . . . . . . 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 594	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ 𝑠 ∈ 𝐶) → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉))
127	126	ex 412	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑠 ∈ 𝐶 → (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉)))
128		eleq1 2816	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑔 → (𝐹 ∈ 𝑇 ↔ 𝑔 ∈ 𝑇))
129		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑔 → (𝑅‘𝐹) = (𝑅‘𝑔))
130	129	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑔 → ((𝑅‘𝐹) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
131	128, 130	anbi12d 632	. . . . . . . . . 10 ⊢ (𝐹 = 𝑔 → ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑉) ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)))
132	131	biimpcd 249	. . . . . . . . 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 3132	. . . . . . 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 212	. . . . . 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 282	. . . . 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 6858	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑅‘𝑓) = (𝑅‘𝑔))
138	137	breq1d 5117	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑅‘𝑓) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
139	138	elrab 3659	. . . . 5 ⊢ (𝑔 ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑉} ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉))
140	136, 139	bitr4di 289	. . . 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 41026	. . . . . 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 2814	. . . 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 282	. . 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 283	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → (𝑔 ∈ ran 𝐺 ↔ 𝑔 ∈ (𝐼‘𝑉)))
152	151	eqrdv 2727	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405   class class class wbr 5107   ↦ cmpt 5188  ran crn 5639   Fn wfn 6506  ‘cfv 6511  ℩crio 7343  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  Latclat 18390  Atomscatm 39256  HLchlt 39343  LHypclh 39978  LTrncltrn 40095  trLctrl 40152  DIsoAcdia 41022

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-riotaBAD 38946

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-undef 8252  df-map 8801  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492  df-lplanes 39493  df-lvols 39494  df-lines 39495  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982  df-laut 39983  df-ldil 40098  df-ltrn 40099  df-trl 40153  df-disoa 41023

This theorem is referenced by: (None)