Theorem pmodl42N 37891

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 1189	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝐾 ∈ HL)
2		simpl3 1191	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ∈ 𝑆)
3		eqid 2733	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmodl42.s	. . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	psubssat 37794	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
6	1, 2, 5	syl2anc 583	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ⊆ (Atoms‘𝐾))
7		simpl2 1190	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ∈ 𝑆)
8	3, 4	psubssat 37794	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
9	1, 7, 8	syl2anc 583	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (Atoms‘𝐾))
10		simprl 767	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑍 ∈ 𝑆)
11	3, 4	psubssat 37794	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) → 𝑍 ⊆ (Atoms‘𝐾))
12	1, 10, 11	syl2anc 583	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑍 ⊆ (Atoms‘𝐾))
13		pmodl42.p	. . . . . . 7 ⊢ + = (+𝑃‘𝐾)
14	3, 13	paddssat 37854	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
15	1, 9, 12, 14	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
16		simprr 769	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑊 ∈ 𝑆)
17	4, 13	paddclN 37882	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆) → (𝑌 + 𝑊) ∈ 𝑆)
18	1, 2, 16, 17	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + 𝑊) ∈ 𝑆)
19	3, 4	psubssat 37794	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑆) → 𝑊 ⊆ (Atoms‘𝐾))
20	1, 16, 19	syl2anc 583	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑊 ⊆ (Atoms‘𝐾))
21	3, 13	sspadd1 37855	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾)) → 𝑌 ⊆ (𝑌 + 𝑊))
22	1, 6, 20, 21	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ⊆ (𝑌 + 𝑊))
23	3, 4, 13	pmod1i 37888	. . . . . 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 1115	. . . . 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 1381	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
26		incom 4138	. . . 4 ⊢ ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))
27	25, 26	eqtr3di 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))
28	27	oveq2d 7311	. 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 4174	. . . 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 37878	. . 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 1370	. 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 37878	. . . . . . 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 1370	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
35	3, 13	padd12N 37879	. . . . . . 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 1370	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
37	34, 36	eqtrd 2773	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
38	3, 13	paddass 37878	. . . . . 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 1370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
40	37, 39	ineq12d 4150	. . . 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 4138	. . . 4 ⊢ ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍)))
42	40, 41	eqtrdi 2789	. . 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 37794	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑌 + 𝑊) ∈ 𝑆) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
44	1, 18, 43	syl2anc 583	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
45	4, 13	paddclN 37882	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑍 ∈ 𝑆) → (𝑋 + 𝑍) ∈ 𝑆)
46	1, 7, 10, 45	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + 𝑍) ∈ 𝑆)
47	4, 13	paddclN 37882	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆 ∧ (𝑋 + 𝑍) ∈ 𝑆) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
48	1, 2, 46, 47	syl3anc 1369	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
49	3, 13	sspadd1 37855	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑍))
50	1, 9, 12, 49	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (𝑋 + 𝑍))
51	3, 13	sspadd2 37856	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
52	1, 15, 6, 51	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
53	50, 52	sstrd 3933	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)))
54	3, 4, 13	pmod1i 37888	. . . . 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 1115	. . . 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 1381	. . 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 2773	. 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 2784	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 395   ∧ w3a 1085   = wceq 1537   ∈ wcel 2101   ∩ cin 3888   ⊆ wss 3889  ‘cfv 6447  (class class class)co 7295  Atomscatm 37303  HLchlt 37390  PSubSpcpsubsp 37536  +_𝑃cpadd 37835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-1st 7851  df-2nd 7852  df-proset 18041  df-poset 18059  df-plt 18076  df-lub 18092  df-glb 18093  df-join 18094  df-meet 18095  df-p0 18171  df-lat 18178  df-clat 18245  df-oposet 37216  df-ol 37218  df-oml 37219  df-covers 37306  df-ats 37307  df-atl 37338  df-cvlat 37362  df-hlat 37391  df-psubsp 37543  df-padd 37836

This theorem is referenced by:  pl42lem4N  38022