Theorem cvrat4 37906

Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 31339 analog.) (Contributed by NM, 30-Nov-2011.)

Hypotheses

Ref	Expression
cvrat4.b	⊢ 𝐵 = (Base‘𝐾)
cvrat4.l	⊢ ≤ = (le‘𝐾)
cvrat4.j	⊢ ∨ = (join‘𝐾)
cvrat4.z	⊢ 0 = (0.‘𝐾)
cvrat4.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cvrat4	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))

Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   ∨ ,𝑟   𝐾,𝑟   ≤ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑋,𝑟

Allowed substitution hint:   0 (𝑟)

Proof of Theorem cvrat4

Step	Hyp	Ref	Expression
1		hlatl 37822	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	adantr 481	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
3		simpr1 1194	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
4		cvrat4.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐾)
5		cvrat4.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
6		cvrat4.z	. . . . . . . . . . 11 ⊢ 0 = (0.‘𝐾)
7		cvrat4.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Atoms‘𝐾)
8	4, 5, 6, 7	atlex 37778	. . . . . . . . . 10 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)
9	8	3exp 1119	. . . . . . . . 9 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋)))
10	2, 3, 9	sylc 65	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
11	10	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋))
12		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝐾 ∈ HL)
13		simplr3 1217	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ∈ 𝐴)
14		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
15		cvrat4.j	. . . . . . . . . . . . . . 15 ⊢ ∨ = (join‘𝐾)
16	5, 15, 7	hlatlej1 37837	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
17	12, 13, 14, 16	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑟))
18		breq1 5108	. . . . . . . . . . . . 13 ⊢ (𝑃 = 𝑄 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑄 ≤ (𝑄 ∨ 𝑟)))
19	17, 18	syl5ibr 245	. . . . . . . . . . . 12 ⊢ (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑟 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑟)))
20	19	expd 416	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟))))
21	20	impcom 408	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → 𝑃 ≤ (𝑄 ∨ 𝑟)))
22	21	anim2d 612	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 ≤ 𝑋 ∧ 𝑟 ∈ 𝐴) → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
23	22	expcomd 417	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑟 ∈ 𝐴 → (𝑟 ≤ 𝑋 → (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
24	23	reximdvai 3162	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (∃𝑟 ∈ 𝐴 𝑟 ≤ 𝑋 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
25	11, 24	syld 47	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
26	25	ex 413	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
27	26	a1i 11	. . . 4 ⊢ (𝑃 ≤ (𝑋 ∨ 𝑄) → ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → (𝑋 ≠ 0 → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))))
28	27	com4l 92	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → (𝑋 ≠ 0 → (𝑃 ≤ (𝑋 ∨ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))))
29	28	imp4a 423	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 = 𝑄 → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
30		hllat 37825	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
31	30	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
32		simpr3 1196	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
33	4, 7	atbase 37751	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
34	32, 33	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
35	4, 5, 15	latleeqj2 18341	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
36	31, 34, 3, 35	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
37	36	biimpa 477	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = 𝑋)
38	37	breq2d 5117	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ 𝑃 ≤ 𝑋))
39	38	biimpa 477	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋)
40	39	expl 458	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ 𝑋))
41		simpl 483	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
42		simpr2 1195	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
43	5, 15, 7	hlatlej2 37838	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → 𝑃 ≤ (𝑄 ∨ 𝑃))
44	41, 32, 42, 43	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ≤ (𝑄 ∨ 𝑃))
45	40, 44	jctird 527	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
46	45, 42	jctild 526	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃)))))
47	46	impl 456	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
48		breq1 5108	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑟 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
49		oveq2 7365	. . . . . . . 8 ⊢ (𝑟 = 𝑃 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑃))
50	49	breq2d 5117	. . . . . . 7 ⊢ (𝑟 = 𝑃 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑃)))
51	48, 50	anbi12d 631	. . . . . 6 ⊢ (𝑟 = 𝑃 → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))))
52	51	rspcev 3581	. . . . 5 ⊢ ((𝑃 ∈ 𝐴 ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑃))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
53	47, 52	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
54	53	adantrl 714	. . 3 ⊢ ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≤ 𝑋) ∧ (𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
55	54	exp31 420	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ≤ 𝑋 → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
56		simpr 485	. . 3 ⊢ ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
57		ioran 982	. . . . 5 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
58		df-ne 2944	. . . . . 6 ⊢ (𝑃 ≠ 𝑄 ↔ ¬ 𝑃 = 𝑄)
59	58	anbi1i 624	. . . . 5 ⊢ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
60	57, 59	bitr4i 277	. . . 4 ⊢ (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) ↔ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋))
61		eqid 2736	. . . . . . . . . 10 ⊢ (meet‘𝐾) = (meet‘𝐾)
62	4, 5, 15, 61, 7	cvrat3 37905	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴))
63	62	3expd 1353	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ≠ 𝑄 → (¬ 𝑄 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴))))
64	63	imp4c 424	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴))
65	4, 7	atbase 37751	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
66	42, 65	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
67	4, 15	latjcl 18328	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
68	31, 66, 34, 67	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
69	4, 5, 61	latmle1 18353	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
70	31, 3, 68, 69	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
71	70	adantr 481	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋)
72		simpll 765	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝐾 ∈ HL)
73	63	imp44 429	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
74		simplr2 1216	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
75	34	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑄 ∈ 𝐵)
76	73, 74, 75	3jca 1128	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵))
77	72, 76	jca 512	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)))
78	4, 5, 61, 6, 7	atnle 37779	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
79	2, 32, 3, 78	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
80	4, 61	latmcom 18352	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
81	31, 34, 3, 80	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
82	81	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
83	79, 82	bitrd 278	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
84	4, 61	latmcl 18329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
85	31, 3, 68, 84	syl3anc 1371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
86	85, 3, 34	3jca 1128	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵))
87	31, 86	jca 512	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵)))
88	4, 5, 61	latmlem2 18359	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑄(meet‘𝐾)𝑋)))
89	87, 70, 88	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑄(meet‘𝐾)𝑋))
90	89, 81	breqtrd 5131	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄))
91		breq2 5109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋(meet‘𝐾)𝑄) = 0 → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ (𝑋(meet‘𝐾)𝑄) ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ))
92	90, 91	syl5ibcom 244	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ))
93		hlop 37824	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	93	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ OP)
95	4, 61	latmcl 18329	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
96	31, 34, 85, 95	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵)
97	4, 5, 6	ople0 37649	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OP ∧ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ∈ 𝐵) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
98	94, 96, 97	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) ≤ 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
99	92, 98	sylibd 238	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
100	83, 99	sylbid 239	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ 𝑄 ≤ 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ))
101	100	imp 407	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
102	101	adantrl 714	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
103	102	adantrr 715	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 )
104	4, 5, 61	latmle2 18354	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
105	31, 3, 68, 104	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
106	4, 15	latjcom 18336	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
107	31, 66, 34, 106	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
108	105, 107	breqtrd 5131	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃))
109	108	adantr 481	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃))
110	30	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝐾 ∈ Lat)
111		simpr3 1196	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → 𝑄 ∈ 𝐵)
112		simpr1 1194	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴)
113	4, 7	atbase 37751	. . . . . . . . . . . . . 14 ⊢ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
114	112, 113	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵)
115	4, 61	latmcom 18352	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄))
116	110, 111, 114, 115	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄))
117	116	eqeq1d 2738	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 ↔ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄) = 0 ))
118	4, 5, 15, 61, 6, 7	hlexch3 37854	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄) = 0 ) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
119	118	3expia 1121	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → (((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))(meet‘𝐾)𝑄) = 0 → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
120	117, 119	sylbid 239	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))) = 0 → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ (𝑄 ∨ 𝑃) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
121	77, 103, 109, 120	syl3c 66	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
122	71, 121	jca 512	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
123	122	ex 413	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
124	64, 123	jcad 513	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))))
125		breq1 5108	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑟 ≤ 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋))
126		oveq2 7365	. . . . . . . . 9 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑄 ∨ 𝑟) = (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))
127	126	breq2d 5117	. . . . . . . 8 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)))))
128	125, 127	anbi12d 631	. . . . . . 7 ⊢ (𝑟 = (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) → ((𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)) ↔ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))))
129	128	rspcev 3581	. . . . . 6 ⊢ (((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝑋(meet‘𝐾)(𝑃 ∨ 𝑄)) ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ (𝑋(meet‘𝐾)(𝑃 ∨ 𝑄))))) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))
130	124, 129	syl6 35	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))
131	130	expd 416	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
132	60, 131	biimtrid 241	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
133	56, 132	syl7 74	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (¬ (𝑃 = 𝑄 ∨ 𝑄 ≤ 𝑋) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟)))))
134	29, 55, 133	ecase3d 1032	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ∃𝑟 ∈ 𝐴 (𝑟 ≤ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑟))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  0.cp0 18312  Latclat 18320  OPcops 37634  Atomscatm 37725  AtLatcal 37726  HLchlt 37812

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813

This theorem is referenced by:  cvrat42  37907  ps-2  37941