Theorem cvrat3 39702

Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32471 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 2736	. . . . . . . . . . . 12 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
5		cvrat3.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	cvr1 39670	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
7	6	3adant3r2 1184	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
8	7	biimpa 476	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄))
9	8	adantrr 717	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄))
10		hllat 39623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
11	10	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
12		simpr2 1196	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
13	1, 5	atbase 39549	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
14	12, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
15		simpr3 1197	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
16	1, 5	atbase 39549	. . . . . . . . . . . . . . . . . 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 18370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
19	11, 14, 17, 18	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
20	19	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
21		simpr1 1195	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
22	1, 3	latjass 18406	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
23	11, 21, 17, 14, 22	syl13anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
24	20, 23	eqtr4d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
25	24	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
26	1, 3	latjcl 18362	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
27	11, 21, 17, 26	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
28	1, 2, 3	latjlej2 18377	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
30	29	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ∨ 𝑄) ∨ 𝑃) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
31	25, 30	eqbrtrd 5120	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
32	1, 3	latjidm 18385	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑄) ∈ 𝐵) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
33	11, 27, 32	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
34	33	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
35	31, 34	breqtrd 5124	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ≤ (𝑋 ∨ 𝑄))
36		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
37	2, 3, 5	hlatlej2 39636	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
38	36, 12, 15, 37	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ≤ (𝑃 ∨ 𝑄))
39	1, 3	latjcl 18362	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
40	11, 14, 17, 39	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
41	1, 2, 3	latjlej2 18377	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ (𝑃 ∨ 𝑄)))
45	1, 3	latjcl 18362	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
46	11, 21, 40, 45	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
47	1, 2	latasymb 18365	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 712	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
51	50	breq2d 5110	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄)) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ 𝑄)))
52	51	adantrl 716	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄)))
54	53	ex 412	. . . . . 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 39680	. . . . . . 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 1373	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄) ↔ 𝑋( ⋖ ‘𝐾)(𝑋 ∨ (𝑃 ∨ 𝑄))))
58	54, 57	sylibrd 259	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)))
59	58	adantr 480	. . . 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 18363	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
61	11, 21, 40, 60	syl3anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐵)
62	1, 3, 4, 5	cvrat2 39689	. . . . . . 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 1121	. . . . . 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 1374	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( ⋖ ‘𝐾)(𝑃 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
65	64	expdimp 452	. . . 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 430	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≠ 𝑄 → (¬ 𝑄 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))))
68	67	3impd 1349	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  meetcmee 18235  Latclat 18354   ⋖ ccvr 39522  Atomscatm 39523  HLchlt 39610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-lat 18355  df-clat 18422  df-oposet 39436  df-ol 39438  df-oml 39439  df-covers 39526  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611

This theorem is referenced by:  cvrat4  39703  2atjm  39705  1cvrat  39736  2llnma1b  40046