Theorem cvrat3 40758

Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 33044 analog.) (Contributed by NM, 30-Nov-2011.)

Hypotheses

Ref	Expression
cvrat3.b	⊢ 𝐵 = (Base‘𝐾)
cvrat3.l	⊢ ≤ = (le‘𝐾)
cvrat3.j	⊢ ∨ = (join‘𝐾)
cvrat3.m	⊢ ∧ = (meet‘𝐾)
cvrat3.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cvrat3	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))

Proof of Theorem cvrat3

Step	Hyp	Ref	Expression
1		cvrat3.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐾)
2		cvrat3.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
3		cvrat3.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
4		eqid 2763	. . . . . . . . . . . 12 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5		cvrat3.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	cvr1 40726	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
7	6	3adant3r2 1202	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
8	7	biimpa 482	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄))
9	8	adantrr 730	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄))
10		hllat 40679	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
11	10	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
12		simpr2 1214	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
13	1, 5	atbase 40605	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
14	12, 13	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
15		simpr3 1215	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
16	1, 5	atbase 40605	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
17	15, 16	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
18	1, 3	latjcom 18662	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
19	11, 14, 17, 18	syl3anc 1399	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
20	19	oveq2d 7465	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
21		simpr1 1213	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
22	1, 3	latjass 18698	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
23	11, 21, 17, 14, 22	syl13anc 1400	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
24	20, 23	eqtr4d 2801	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
25	24	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
26	1, 3	latjcl 18654	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
27	11, 21, 17, 26	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
28	1, 2, 3	latjlej2 18669	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵)) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
29	11, 14, 27, 27, 28	syl13anc 1400	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
30	29	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
31	25, 30	eqbrtrd 5131	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
32	1, 3	latjidm 18677	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑄) ∈ 𝐵) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
33	11, 27, 32	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
34	33	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
35	31, 34	breqtrd 5135	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ (𝑋 ∨ 𝑄))
36		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
37	2, 3, 5	hlatlej2 40692	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
38	36, 12, 15, 37	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ≤ (𝑃 ∨ 𝑄))
39	1, 3	latjcl 18654	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
40	11, 14, 17, 39	syl3anc 1399	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
41	1, 2, 3	latjlej2 18669	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑄 ≤ (𝑃 ∨ 𝑄) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄))))
42	11, 17, 40, 21, 41	syl13anc 1400	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑄) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄))))
43	38, 42	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄)))
44	43	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄)))
45	1, 3	latjcl 18654	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
46	11, 21, 40, 45	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
47	1, 2	latasymb 18657	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵) → (((𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄))) ↔ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄)))
48	11, 46, 27, 47	syl3anc 1399	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄))) ↔ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄)))
49	48	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (((𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄))) ↔ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄)))
50	35, 44, 49	mpbi2and 725	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
51	50	breq2d 5118	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
52	51	adantrl 729	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
53	9, 52	mpbird 260	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄)))
54	53	ex 418	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄))))
55		cvrat3.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
56	1, 3, 55, 4	cvrexch 40736	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → ((𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄))))
57	36, 21, 40, 56	syl3anc 1399	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄))))
58	54, 57	sylibrd 262	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)))
59	58	adantr 486	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)))
60	1, 55	latmcl 18655	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
61	11, 21, 40, 60	syl3anc 1399	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
62	1, 3, 4, 5	cvrat2 40745	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄))) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
63	62	3expia 1139	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
64	36, 61, 12, 15, 63	syl13anc 1400	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
65	64	expdimp 458	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
66	59, 65	syld 48	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
67	66	exp4b 436	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≠ 𝑄 → (¬ 𝑄 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))))
68	67	3impd 1367	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ‘cfv 6564  (class class class)co 7449  Basecbs 17428  lecple 17476  joincjn 18526  meetcmee 18527  Latclat 18646   ⋖ ccvr 40578  Atomscatm 40579  HLchlt 40666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-proset 18509  df-poset 18528  df-plt 18543  df-lub 18559  df-glb 18560  df-join 18561  df-meet 18562  df-p0 18638  df-lat 18647  df-clat 18714  df-oposet 40492  df-ol 40494  df-oml 40495  df-covers 40582  df-ats 40583  df-atl 40614  df-cvlat 40638  df-hlat 40667

This theorem is used by:  cvrat4  40759  2atjm  40761  1cvrat  40792  2llnma1b  41102