Theorem cvlexchb1 36498

Description: An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
cvlexchb1	⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))

Proof of Theorem cvlexchb1

Step	Hyp	Ref	Expression
1		cvllat 36494	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2	1	adantr 483	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ Lat)
3		simpr3 1192	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1191	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑄 ∈ 𝐴)
5		cvlexch.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
6		cvlexch.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	atbase 36457	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
8	4, 7	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑄 ∈ 𝐵)
9		cvlexch.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
10		cvlexch.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
11	5, 9, 10	latlej1 17653	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄))
12	2, 3, 8, 11	syl3anc 1367	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
13	12	3adant3 1128	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑄))
14	13	adantr 483	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
15		simpr 487	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
16		simpr1 1190	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴)
17	5, 6	atbase 36457	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐵)
19	5, 10	latjcl 17644	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
20	2, 3, 8, 19	syl3anc 1367	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
21	5, 9, 10	latjle12 17655	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
22	2, 3, 18, 20, 21	syl13anc 1368	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
23	22	3adant3 1128	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
24	23	adantr 483	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
25	14, 15, 24	mpbi2and 710	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄))
26	5, 9, 10	latlej1 17653	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑃))
27	2, 3, 18, 26	syl3anc 1367	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
28	27	3adant3 1128	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑃))
29	28	adantr 483	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
30	5, 9, 10, 6	cvlexch1 36496	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
31	30	imp 409	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))
32	5, 10	latjcl 17644	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) ∈ 𝐵)
33	2, 3, 18, 32	syl3anc 1367	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑃) ∈ 𝐵)
34	5, 9, 10	latjle12 17655	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ (𝑋 ∨ 𝑃) ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
35	2, 3, 8, 33, 34	syl13anc 1368	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
36	35	3adant3 1128	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
37	36	adantr 483	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
38	29, 31, 37	mpbi2and 710	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃))
39	5, 9	latasymb 17647	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑃) ∈ 𝐵 ∧ (𝑋 ∨ 𝑄) ∈ 𝐵) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
40	2, 33, 20, 39	syl3anc 1367	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
41	40	3adant3 1128	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
42	41	adantr 483	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
43	25, 38, 42	mpbi2and 710	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))
44	43	ex 415	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
45	5, 9, 10	latlej2 17654	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑃 ≤ (𝑋 ∨ 𝑃))
46	2, 3, 18, 45	syl3anc 1367	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ≤ (𝑋 ∨ 𝑃))
47		breq2 5056	. . . 4 ⊢ ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → (𝑃 ≤ (𝑋 ∨ 𝑃) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
48	46, 47	syl5ibcom 247	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
49	48	3adant3 1128	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
50	44, 49	impbid 214	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5052  ‘cfv 6341  (class class class)co 7142  Basecbs 16466  lecple 16555  joincjn 17537  Latclat 17638  Atomscatm 36431  CvLatclc 36433

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  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 7100  df-ov 7145  df-oprab 7146  df-proset 17521  df-poset 17539  df-lub 17567  df-glb 17568  df-join 17569  df-meet 17570  df-lat 17639  df-ats 36435  df-atl 36466  df-cvlat 36490

This theorem is referenced by:  cvlexchb2  36499  cvlexch4N  36501  cvlatexchb1  36502  cvlcvr1  36507  hlexchb1  36552