osumcllem11N - Mathbox for Norm Megill

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Theorem osumcllem11N 39140

Description: Lemma for osumclN 39141. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
osumcl.p	⊢ + = (+_𝑃‘𝐾)
osumcl.o	⊢ ⊥ = (⊥_𝑃‘𝐾)
osumcl.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
osumcllem11N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))

Proof of Theorem osumcllem11N Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nonconne 2950	. 2 ⊢ ¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)
2		simpl1 1189	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝐾 ∈ HL)
3		simpl2 1190	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑋 ∈ 𝐶)
4		eqid 2730	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		osumcl.c	. . . . . . . 8 ⊢ 𝐶 = (PSubCl‘𝐾)
6	4, 5	psubclssatN 39115	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7	2, 3, 6	syl2anc 582	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ (Atoms‘𝐾))
8		simpl3 1191	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑌 ∈ 𝐶)
9	4, 5	psubclssatN 39115	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → 𝑌 ⊆ (Atoms‘𝐾))
10	2, 8, 9	syl2anc 582	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → 𝑌 ⊆ (Atoms‘𝐾))
11		osumcl.p	. . . . . . 7 ⊢ + = (+_𝑃‘𝐾)
12	4, 11	paddssat 38988	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
13	2, 7, 10, 12	syl3anc 1369	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
14		osumcl.o	. . . . . 6 ⊢ ⊥ = (⊥_𝑃‘𝐾)
15	4, 14	2polssN 39089	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))
16	2, 13, 15	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))
17		df-pss 3966	. . . . . . 7 ⊢ ((𝑋 + 𝑌) ⊊ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ↔ ((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))))
18		pssnel 4469	. . . . . . 7 ⊢ ((𝑋 + 𝑌) ⊊ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) → ∃𝑝(𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)))
19	17, 18	sylbir 234	. . . . . 6 ⊢ (((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) → ∃𝑝(𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)))
20		df-rex 3069	. . . . . 6 ⊢ (∃𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ¬ 𝑝 ∈ (𝑋 + 𝑌) ↔ ∃𝑝(𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)))
21	19, 20	sylibr 233	. . . . 5 ⊢ (((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) → ∃𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ¬ 𝑝 ∈ (𝑋 + 𝑌))
22		eqid 2730	. . . . . . . . . . 11 ⊢ (le‘𝐾) = (le‘𝐾)
23		eqid 2730	. . . . . . . . . . 11 ⊢ (join‘𝐾) = (join‘𝐾)
24		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑋 + {𝑝}) = (𝑋 + {𝑝})
25		eqid 2730	. . . . . . . . . . 11 ⊢ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))
26	22, 23, 4, 11, 14, 5, 24, 25	osumcllem9N 39138	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) = 𝑋)
27		simp11 1201	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL)
28		simp12 1202	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ∈ 𝐶)
29	27, 28, 6	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (Atoms‘𝐾))
30		simp13 1203	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌 ∈ 𝐶)
31	27, 30, 9	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌 ⊆ (Atoms‘𝐾))
32	13	3adantr3 1169	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
33	32	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + 𝑌) ⊆ (Atoms‘𝐾))
34	4, 14	polssatN 39082	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ (Atoms‘𝐾)) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ (Atoms‘𝐾))
35	27, 33, 34	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ (Atoms‘𝐾))
36	4, 14	polssatN 39082	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘(𝑋 + 𝑌)) ⊆ (Atoms‘𝐾)) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ (Atoms‘𝐾))
37	27, 35, 36	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ (Atoms‘𝐾))
38		simp23 1206	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))
39	37, 38	sseldd 3982	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
40		simp3 1136	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌))
41	22, 23, 4, 11, 14, 5, 24, 25	osumcllem10N 39139	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ≠ 𝑋)
42	27, 29, 31, 39, 40, 41	syl311anc 1382	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ≠ 𝑋)
43	26, 42	pm2.21ddne 3024	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))
44	43	3exp 1117	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → ((𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) → (¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))))
45	44	3expd 1351	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → (𝑋 ⊆ ( ⊥ ‘𝑌) → (𝑋 ≠ ∅ → (𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) → (¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))))))
46	45	imp32 417	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) → (¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))))
47	46	rexlimdv 3151	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (∃𝑝 ∈ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ¬ 𝑝 ∈ (𝑋 + 𝑌) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
48	21, 47	syl5 34	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (((𝑋 + 𝑌) ⊆ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∧ (𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
49	16, 48	mpand 691	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → ((𝑋 + 𝑌) ≠ ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
50	49	necon1bd 2956	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))))
51	1, 50	mpi 20	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 ⊆ wss 3947 ⊊ wpss 3948 ∅c0 4321 {csn 4627 ‘cfv 6542 (class class class)co 7411 lecple 17208 joincjn 18268 Atomscatm 38436 HLchlt 38523 +_𝑃cpadd 38969 ⊥_𝑃cpolN 39076 PSubClcpscN 39108

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-polarityN 39077 df-psubclN 39109

This theorem is referenced by: osumclN 39141

W3C validator