Theorem pmodl42N 37551

Description: Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pmodl42.s	⊢ 𝑆 = (PSubSp‘𝐾)
pmodl42.p	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
pmodl42N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))

Proof of Theorem pmodl42N

Step	Hyp	Ref	Expression
1		simpl1 1193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝐾 ∈ HL)
2		simpl3 1195	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ∈ 𝑆)
3		eqid 2736	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmodl42.s	. . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	psubssat 37454	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
6	1, 2, 5	syl2anc 587	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ⊆ (Atoms‘𝐾))
7		simpl2 1194	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ∈ 𝑆)
8	3, 4	psubssat 37454	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
9	1, 7, 8	syl2anc 587	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (Atoms‘𝐾))
10		simprl 771	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑍 ∈ 𝑆)
11	3, 4	psubssat 37454	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) → 𝑍 ⊆ (Atoms‘𝐾))
12	1, 10, 11	syl2anc 587	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑍 ⊆ (Atoms‘𝐾))
13		pmodl42.p	. . . . . . 7 ⊢ + = (+𝑃‘𝐾)
14	3, 13	paddssat 37514	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
15	1, 9, 12, 14	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
16		simprr 773	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑊 ∈ 𝑆)
17	4, 13	paddclN 37542	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆) → (𝑌 + 𝑊) ∈ 𝑆)
18	1, 2, 16, 17	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + 𝑊) ∈ 𝑆)
19	3, 4	psubssat 37454	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑆) → 𝑊 ⊆ (Atoms‘𝐾))
20	1, 16, 19	syl2anc 587	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑊 ⊆ (Atoms‘𝐾))
21	3, 13	sspadd1 37515	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾)) → 𝑌 ⊆ (𝑌 + 𝑊))
22	1, 6, 20, 21	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ⊆ (𝑌 + 𝑊))
23	3, 4, 13	pmod1i 37548	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ∈ 𝑆)) → (𝑌 ⊆ (𝑌 + 𝑊) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
24	23	3impia 1119	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ∈ 𝑆) ∧ 𝑌 ⊆ (𝑌 + 𝑊)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
25	1, 6, 15, 18, 22, 24	syl131anc 1385	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
26		incom 4101	. . . 4 ⊢ ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))
27	25, 26	eqtr3di 2786	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))
28	27	oveq2d 7207	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
29		ssinss1 4138	. . . 4 ⊢ ((𝑋 + 𝑍) ⊆ (Atoms‘𝐾) → ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)) ⊆ (Atoms‘𝐾))
30	15, 29	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)) ⊆ (Atoms‘𝐾))
31	3, 13	paddass 37538	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)) ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
32	1, 9, 6, 30, 31	syl13anc 1374	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
33	3, 13	paddass 37538	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
34	1, 9, 6, 12, 33	syl13anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
35	3, 13	padd12N 37539	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
36	1, 9, 6, 12, 35	syl13anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
37	34, 36	eqtrd 2771	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
38	3, 13	paddass 37538	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
39	1, 9, 6, 20, 38	syl13anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
40	37, 39	ineq12d 4114	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))))
41		incom 4101	. . . 4 ⊢ ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍)))
42	40, 41	eqtrdi 2787	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))))
43	3, 4	psubssat 37454	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑌 + 𝑊) ∈ 𝑆) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
44	1, 18, 43	syl2anc 587	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
45	4, 13	paddclN 37542	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆) → (𝑋 + 𝑍) ∈ 𝑆)
46	1, 7, 10, 45	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + 𝑍) ∈ 𝑆)
47	4, 13	paddclN 37542	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆 ∧ (𝑋 + 𝑍) ∈ 𝑆) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
48	1, 2, 46, 47	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
49	3, 13	sspadd1 37515	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑍))
50	1, 9, 12, 49	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (𝑋 + 𝑍))
51	3, 13	sspadd2 37516	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
52	1, 15, 6, 51	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
53	50, 52	sstrd 3897	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)))
54	3, 4, 13	pmod1i 37548	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ⊆ (Atoms‘𝐾) ∧ (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)) → (𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))))
55	54	3impia 1119	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ⊆ (Atoms‘𝐾) ∧ (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆) ∧ 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍))) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
56	1, 9, 44, 48, 53, 55	syl131anc 1385	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
57	42, 56	eqtrd 2771	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
58	28, 32, 57	3eqtr4rd 2782	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ∩ cin 3852   ⊆ wss 3853  ‘cfv 6358  (class class class)co 7191  Atomscatm 36963  HLchlt 37050  PSubSpcpsubsp 37196  +_𝑃cpadd 37495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-proset 17756  df-poset 17774  df-plt 17790  df-lub 17806  df-glb 17807  df-join 17808  df-meet 17809  df-p0 17885  df-lat 17892  df-clat 17959  df-oposet 36876  df-ol 36878  df-oml 36879  df-covers 36966  df-ats 36967  df-atl 36998  df-cvlat 37022  df-hlat 37051  df-psubsp 37203  df-padd 37496

This theorem is referenced by:  pl42lem4N  37682