Theorem cvlexchb1 40653

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 40649	. . . . . . . . 9 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
2	1	adantr 486	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ Lat)
3		simpr3 1215	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
4		simpr2 1214	. . . . . . . . 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 40612	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
8	4, 7	syl 18	. . . . . . . 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 18672	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄))
12	2, 3, 8, 11	syl3anc 1399	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
13	12	3adant3 1150	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑄))
14	13	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
15		simpr 490	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
16		simpr1 1213	. . . . . . . . 9 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴)
17	5, 6	atbase 40612	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
18	16, 17	syl 18	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐵)
19	5, 10	latjcl 18663	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
20	2, 3, 8, 19	syl3anc 1399	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
21	5, 9, 10	latjle12 18674	. . . . . . . 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 1400	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
23	22	3adant3 1150	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑄) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) ↔ (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄)))
24	23	adantr 486	. . . . 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 725	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄))
26	5, 9, 10	latlej1 18672	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑃))
27	2, 3, 18, 26	syl3anc 1399	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
28	27	3adant3 1150	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ≤ (𝑋 ∨ 𝑃))
29	28	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑋 ≤ (𝑋 ∨ 𝑃))
30	5, 9, 10, 6	cvlexch1 40651	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
31	30	imp 412	. . . . 5 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))
32	5, 10	latjcl 18663	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑋 ∨ 𝑃) ∈ 𝐵)
33	2, 3, 18, 32	syl3anc 1399	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ∨ 𝑃) ∈ 𝐵)
34	5, 9, 10	latjle12 18674	. . . . . . . 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 1400	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
36	35	3adant3 1150	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ≤ (𝑋 ∨ 𝑃) ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)))
37	36	adantr 486	. . . . 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 725	. . . 4 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃))
39	5, 9	latasymb 18666	. . . . . . 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 1399	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (((𝑋 ∨ 𝑃) ≤ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≤ (𝑋 ∨ 𝑃)) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
41	40	3adant3 1150	. . . . 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 486	. . . 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 725	. . 3 ⊢ (((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄))
44	43	ex 418	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))
45	5, 9, 10	latlej2 18673	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → 𝑃 ≤ (𝑋 ∨ 𝑃))
46	2, 3, 18, 45	syl3anc 1399	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ≤ (𝑋 ∨ 𝑃))
47		breq2 5112	. . . 4 ⊢ ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → (𝑃 ≤ (𝑋 ∨ 𝑃) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
48	46, 47	syl5ibcom 248	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
49	48	3adant3 1150	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → ((𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄) → 𝑃 ≤ (𝑋 ∨ 𝑄)))
50	44, 49	impbid 215	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) ↔ (𝑋 ∨ 𝑃) = (𝑋 ∨ 𝑄)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  lecple 17479  joincjn 18535  Latclat 18655  Atomscatm 40586  CvLatclc 40588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-proset 18518  df-poset 18537  df-lub 18568  df-glb 18569  df-join 18570  df-meet 18571  df-lat 18656  df-ats 40590  df-atl 40621  df-cvlat 40645

This theorem is used by:  cvlexchb2  40654  cvlexch4N  40656  cvlatexchb1  40657  cvlcvr1  40662  hlexchb1  40707