Theorem pmodl42N 40023

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 1192	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝐾 ∈ HL)
2		simpl3 1194	. . . . . 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 39926	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
6	1, 2, 5	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑌 ⊆ (Atoms‘𝐾))
7		simpl2 1193	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ∈ 𝑆)
8	3, 4	psubssat 39926	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
9	1, 7, 8	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (Atoms‘𝐾))
10		simprl 770	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑍 ∈ 𝑆)
11	3, 4	psubssat 39926	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) → 𝑍 ⊆ (Atoms‘𝐾))
12	1, 10, 11	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑍 ⊆ (Atoms‘𝐾))
13		pmodl42.p	. . . . . . 7 ⊢ + = (+𝑃‘𝐾)
14	3, 13	paddssat 39986	. . . . . 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 772	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑊 ∈ 𝑆)
17	4, 13	paddclN 40014	. . . . . 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 39926	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝑆) → 𝑊 ⊆ (Atoms‘𝐾))
20	1, 16, 19	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑊 ⊆ (Atoms‘𝐾))
21	3, 13	sspadd1 39987	. . . . . 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 40020	. . . . . 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 1117	. . . . 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 4158	. . . 4 ⊢ ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))
27	25, 26	eqtr3di 2783	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))
28	27	oveq2d 7371	. 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 4195	. . . 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 40010	. . 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 40010	. . . . . . 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 40011	. . . . . . 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 2768	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
38	3, 13	paddass 40010	. . . . . 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 4170	. . . 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 4158	. . . 4 ⊢ ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍)))
42	40, 41	eqtrdi 2784	. . 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 39926	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑌 + 𝑊) ∈ 𝑆) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
44	1, 18, 43	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
45	4, 13	paddclN 40014	. . . . . 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 40014	. . . . 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 39987	. . . . . 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 39988	. . . . . 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 3941	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)))
54	3, 4, 13	pmod1i 40020	. . . . 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 1117	. . . 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 2768	. 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 2779	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 1086   = wceq 1541   ∈ wcel 2113   ∩ cin 3897   ⊆ wss 3898  ‘cfv 6489  (class class class)co 7355  Atomscatm 39435  HLchlt 39522  PSubSpcpsubsp 39668  +_𝑃cpadd 39967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-proset 18208  df-poset 18227  df-plt 18242  df-lub 18258  df-glb 18259  df-join 18260  df-meet 18261  df-p0 18337  df-lat 18346  df-clat 18413  df-oposet 39348  df-ol 39350  df-oml 39351  df-covers 39438  df-ats 39439  df-atl 39470  df-cvlat 39494  df-hlat 39523  df-psubsp 39675  df-padd 39968

This theorem is referenced by:  pl42lem4N  40154