Theorem cvrat3 37054

Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 30293 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 2759	. . . . . . . . . . . 12 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5		cvrat3.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	cvr1 37022	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
7	6	3adant3r2 1181	. . . . . . . . . 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 36975	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
11	10	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
12		simpr2 1193	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
13	1, 5	atbase 36901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
14	12, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
15		simpr3 1194	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
16	1, 5	atbase 36901	. . . . . . . . . . . . . . . . . 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 17750	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
19	11, 14, 17, 18	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
20	19	oveq2d 7173	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
21		simpr1 1192	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
22	1, 3	latjass 17786	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
23	11, 21, 17, 14, 22	syl13anc 1370	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
24	20, 23	eqtr4d 2797	. . . . . . . . . . . . . 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 17742	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
27	11, 21, 17, 26	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
28	1, 2, 3	latjlej2 17757	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 5059	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
32	1, 3	latjidm 17765	. . . . . . . . . . . . . 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 5063	. . . . . . . . . . 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 36988	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
38	36, 12, 15, 37	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ≤ (𝑃 ∨ 𝑄))
39	1, 3	latjcl 17742	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
40	11, 14, 17, 39	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
41	1, 2, 3	latjlej2 17757	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 17742	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
46	11, 21, 40, 45	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
47	1, 2	latasymb 17745	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 5049	. . . . . . . . 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 37032	. . . . . . 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 1369	. . . . . 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 17743	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
61	11, 21, 40, 60	syl3anc 1369	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
62	1, 3, 4, 5	cvrat2 37041	. . . . . . 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 1119	. . . . . 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 1370	. . . . 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 1346	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 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952   class class class wbr 5037  ‘cfv 6341  (class class class)co 7157  Basecbs 16556  lecple 16645  joincjn 17635  meetcmee 17636  Latclat 17736   ⋖ ccvr 36874  Atomscatm 36875  HLchlt 36962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7115  df-ov 7160  df-oprab 7161  df-proset 17619  df-poset 17637  df-plt 17649  df-lub 17665  df-glb 17666  df-join 17667  df-meet 17668  df-p0 17730  df-lat 17737  df-clat 17799  df-oposet 36788  df-ol 36790  df-oml 36791  df-covers 36878  df-ats 36879  df-atl 36910  df-cvlat 36934  df-hlat 36963

This theorem is referenced by:  cvrat4  37055  2atjm  37057  1cvrat  37088  2llnma1b  37398