Theorem osumcllem9N 35250

Description: Lemma for osumclN 35253. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
osumcllem.l	⊢ ≤ = (le‘𝐾)
osumcllem.j	⊢ ∨ = (join‘𝐾)
osumcllem.a	⊢ 𝐴 = (Atoms‘𝐾)
osumcllem.p	⊢ + = (+𝑃‘𝐾)
osumcllem.o	⊢ ⊥ = (⊥𝑃‘𝐾)
osumcllem.c	⊢ 𝐶 = (PSubCl‘𝐾)
osumcllem.m	⊢ 𝑀 = (𝑋 + {𝑝})
osumcllem.u	⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))

Assertion

Ref	Expression
osumcllem9N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)

Proof of Theorem osumcllem9N

Step	Hyp	Ref	Expression
1		inass 3823	. . . . . . 7 ⊢ ((( ⊥ ‘𝑋) ∩ 𝑈) ∩ 𝑀) = (( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀))
2		simp11 1091	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL)
3		simp13 1093	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌 ∈ 𝐶)
4		simp21 1094	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ ( ⊥ ‘𝑌))
5		osumcllem.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
6		osumcllem.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
7		osumcllem.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
8		osumcllem.p	. . . . . . . . . 10 ⊢ + = (+𝑃‘𝐾)
9		osumcllem.o	. . . . . . . . . 10 ⊢ ⊥ = (⊥𝑃‘𝐾)
10		osumcllem.c	. . . . . . . . . 10 ⊢ 𝐶 = (PSubCl‘𝐾)
11		osumcllem.m	. . . . . . . . . 10 ⊢ 𝑀 = (𝑋 + {𝑝})
12		osumcllem.u	. . . . . . . . . 10 ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))
13	5, 6, 7, 8, 9, 10, 11, 12	osumcllem3N 35244	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘𝑋) ∩ 𝑈) = 𝑌)
14	2, 3, 4, 13	syl3anc 1326	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ⊥ ‘𝑋) ∩ 𝑈) = 𝑌)
15	14	ineq1d 3813	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ((( ⊥ ‘𝑋) ∩ 𝑈) ∩ 𝑀) = (𝑌 ∩ 𝑀))
16	1, 15	syl5eqr 2670	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀)) = (𝑌 ∩ 𝑀))
17		simp12 1092	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ∈ 𝐶)
18	7, 10	psubclssatN 35227	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴)
19	2, 17, 18	syl2anc 693	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ 𝐴)
20	7, 10	psubclssatN 35227	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶) → 𝑌 ⊆ 𝐴)
21	2, 3, 20	syl2anc 693	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌 ⊆ 𝐴)
22		simp22 1095	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ≠ ∅)
23	7, 8	paddssat 35100	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
24	2, 19, 21, 23	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + 𝑌) ⊆ 𝐴)
25	7, 9	polssatN 35194	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴)
26	2, 24, 25	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴)
27	7, 9	polssatN 35194	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ 𝐴)
28	2, 26, 27	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⊆ 𝐴)
29	12, 28	syl5eqss 3649	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈 ⊆ 𝐴)
30		simp23 1096	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ 𝑈)
31	29, 30	sseldd 3604	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ 𝐴)
32		simp3 1063	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌))
33	5, 6, 7, 8, 9, 10, 11, 12	osumcllem8N 35249	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌 ∩ 𝑀) = ∅)
34	2, 19, 21, 4, 22, 31, 32, 33	syl331anc 1351	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌 ∩ 𝑀) = ∅)
35	16, 34	eqtrd 2656	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀)) = ∅)
36	35	fveq2d 6195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀))) = ( ⊥ ‘∅))
37	7, 9	pol0N 35195	. . . . 5 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴)
38	2, 37	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘∅) = 𝐴)
39	36, 38	eqtrd 2656	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘(( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀))) = 𝐴)
40	5, 6, 7, 8, 9, 10, 11, 12	osumcllem1N 35242	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑈 ∩ 𝑀) = 𝑀)
41	2, 19, 21, 30, 40	syl31anc 1329	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈 ∩ 𝑀) = 𝑀)
42	39, 41	ineq12d 3815	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀))) ∩ (𝑈 ∩ 𝑀)) = (𝐴 ∩ 𝑀))
43	7, 9, 10	polsubclN 35238	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∈ 𝐶)
44	2, 26, 43	syl2anc 693	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ∈ 𝐶)
45	12, 44	syl5eqel 2705	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈 ∈ 𝐶)
46	7, 8, 10	paddatclN 35235	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑝 ∈ 𝐴) → (𝑋 + {𝑝}) ∈ 𝐶)
47	2, 17, 31, 46	syl3anc 1326	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ∈ 𝐶)
48	11, 47	syl5eqel 2705	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 ∈ 𝐶)
49	10	psubclinN 35234	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐶 ∧ 𝑀 ∈ 𝐶) → (𝑈 ∩ 𝑀) ∈ 𝐶)
50	2, 45, 48, 49	syl3anc 1326	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈 ∩ 𝑀) ∈ 𝐶)
51	5, 6, 7, 8, 9, 10, 11, 12	osumcllem2N 35243	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑈 ∩ 𝑀))
52	2, 19, 21, 30, 51	syl31anc 1329	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (𝑈 ∩ 𝑀))
53	10, 9	poml6N 35241	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ (𝑈 ∩ 𝑀) ∈ 𝐶) ∧ 𝑋 ⊆ (𝑈 ∩ 𝑀)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀))) ∩ (𝑈 ∩ 𝑀)) = 𝑋)
54	2, 17, 50, 52, 53	syl31anc 1329	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ (𝑈 ∩ 𝑀))) ∩ (𝑈 ∩ 𝑀)) = 𝑋)
55	31	snssd 4340	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → {𝑝} ⊆ 𝐴)
56	7, 8	paddssat 35100	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ {𝑝} ⊆ 𝐴) → (𝑋 + {𝑝}) ⊆ 𝐴)
57	2, 19, 55, 56	syl3anc 1326	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ⊆ 𝐴)
58	11, 57	syl5eqss 3649	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 ⊆ 𝐴)
59		sseqin2 3817	. . 3 ⊢ (𝑀 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑀) = 𝑀)
60	58, 59	sylib 208	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝐴 ∩ 𝑀) = 𝑀)
61	42, 54, 60	3eqtr3rd 2665	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ‘cfv 5888  (class class class)co 6650  lecple 15948  joincjn 16944  Atomscatm 34550  HLchlt 34637  +_𝑃cpadd 35081  ⊥_𝑃cpolN 35188  PSubClcpscN 35220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-polarityN 35189  df-psubclN 35221

This theorem is referenced by:  osumcllem11N  35252