Theorem cdlemm10N 41101

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 7310	. . . . 5 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) ∈ V
2		cdlemm10.g	. . . . 5 ⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞))
3	1, 2	fnmpti 6625	. . . 4 ⊢ 𝐺 Fn 𝐶
4		fvelrnb 6883	. . . 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 7308	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑠 → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠))
8		riotaex 7310	. . . . . . . . . . 11 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) ∈ V
9	7, 2, 8	fvmpt 6930	. . . . . . . . . 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 3151	. . . . . 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 40123	. . . . . . . . . . 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 39347	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ 𝑉)) → 𝐾 ∈ Lat)
27	24, 19	atbase 39272	. . . . . . . . . . . . . 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 40108	. . . . . . . . . . . . 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 18345	. . . . . . . . . . . . 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 39350	. . . . . . . . . . . . 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 18355	. . . . . . . . . . . . 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 40149	. . . . . . . . . . . . . 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 40147	. . . . . . . . . . . . . . . 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 39272	. . . . . . . . . . . . . . . 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 18360	. . . . . . . . . . . . . . 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 5116	. . . . . . . . . . . 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 18352	. . . . . . . . . . 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 40122	. . . . . . . . . . . . 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 5095	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑔‘𝑃) → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ (𝑔‘𝑃) ≤ (𝑃 ∨ 𝑉)))
58		breq1 5095	. . . . . . . . . . . . 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 3651	. . . . . . . . . 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 40568	. . . . . . . . . . 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 7308	. . . . . . . . . . . 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 3577	. . . . . . . . 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 5095	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑠 → (𝑟 ≤ (𝑃 ∨ 𝑉) ↔ 𝑠 ≤ (𝑃 ∨ 𝑉)))
75		breq1 5095	. . . . . . . . . . . . . 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 3651	. . . . . . . . . . 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 40562	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
85	18, 19, 20, 21, 10	ltrniotaval 40564	. . . . . . . . . . . . . 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 40146	. . . . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . 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 39358	. . . . . . . . . . . . . . . . . . 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 39347	. . . . . . . . . . . . . . . . . . 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 39272	. . . . . . . . . . . . . . . . . . . 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 18356	. . . . . . . . . . . . . . . . . . 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 39350	. . . . . . . . . . . . . . . . . . 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 39981	. . . . . . . . . . . . . . . . . . 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 18375	. . . . . . . . . . . . . . . . . 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 40027	. . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . 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 6822	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑔 → (𝑅‘𝐹) = (𝑅‘𝑔))
130	129	breq1d 5102	. . . . . . . . . . 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 3128	. . . . . . 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 6822	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑅‘𝑓) = (𝑅‘𝑔))
138	137	breq1d 5102	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑅‘𝑓) ≤ 𝑉 ↔ (𝑅‘𝑔) ≤ 𝑉))
139	138	elrab 3648	. . . . 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 41015	. . . . . 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 3394   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620   Fn wfn 6477  ‘cfv 6482  ℩crio 7305  (class class class)co 7349  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  Latclat 18337  Atomscatm 39246  HLchlt 39333  LHypclh 39967  LTrncltrn 40084  trLctrl 40141  DIsoAcdia 41011

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 38936

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 39159  df-ol 39161  df-oml 39162  df-covers 39249  df-ats 39250  df-atl 39281  df-cvlat 39305  df-hlat 39334  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-disoa 41012

This theorem is referenced by: (None)