Theorem cvrat3 36738

Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30179 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 2798	. . . . . . . . . . . 12 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5		cvrat3.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	cvr1 36706	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
7	6	3adant3r2 1180	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
8	7	biimpa 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄))
9	8	adantrr 716	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄))
10		hllat 36659	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
11	10	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
12		simpr2 1192	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
13	1, 5	atbase 36585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
14	12, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
15		simpr3 1193	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
16	1, 5	atbase 36585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
17	15, 16	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
18	1, 3	latjcom 17661	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
19	11, 14, 17, 18	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
20	19	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
21		simpr1 1191	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
22	1, 3	latjass 17697	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
23	11, 21, 17, 14, 22	syl13anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
24	20, 23	eqtr4d 2836	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
25	24	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
26	1, 3	latjcl 17653	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
27	11, 21, 17, 26	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
28	1, 2, 3	latjlej2 17668	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
30	29	imp 410	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
31	25, 30	eqbrtrd 5052	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
32	1, 3	latjidm 17676	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑄) ∈ 𝐵) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
33	11, 27, 32	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
34	33	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
35	31, 34	breqtrd 5056	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ (𝑋 ∨ 𝑄))
36		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
37	2, 3, 5	hlatlej2 36672	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
38	36, 12, 15, 37	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ≤ (𝑃 ∨ 𝑄))
39	1, 3	latjcl 17653	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
40	11, 14, 17, 39	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
41	1, 2, 3	latjlej2 17668	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑄) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄))))
43	38, 42	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄)))
44	43	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄)))
45	1, 3	latjcl 17653	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
46	11, 21, 40, 45	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
47	1, 2	latasymb 17656	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 711	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
51	50	breq2d 5042	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
52	51	adantrl 715	. . . . . . . 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 416	. . . . . 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 36716	. . . . . . 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 1368	. . . . . 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 484	. . . 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 17654	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
61	11, 21, 40, 60	syl3anc 1368	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
62	1, 3, 4, 5	cvrat2 36725	. . . . . . 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 1118	. . . . . 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 1369	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
65	64	expdimp 456	. . . 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 47	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
67	66	exp4b 434	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≠ 𝑄 → (¬ 𝑄 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))))
68	67	3impd 1345	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Latclat 17647   ⋖ ccvr 36558  Atomscatm 36559  HLchlt 36646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647

This theorem is referenced by:  cvrat4  36739  2atjm  36741  1cvrat  36772  2llnma1b  37082