Theorem cvlexch2 39317

Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)

Hypotheses

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

Assertion

Ref	Expression
cvlexch2	⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) → 𝑄 ≤ (𝑃 ∨ 𝑋)))

Proof of Theorem cvlexch2

Step	Hyp	Ref	Expression
1		cvlexch.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cvlexch.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cvlexch.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cvlexch.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	cvlexch1 39316	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
6		cvllat 39314	. . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat)
7	6	3ad2ant1 1133	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝐾 ∈ Lat)
8		simp22 1208	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑄 ∈ 𝐴)
9	1, 4	atbase 39277	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
10	8, 9	syl 17	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑄 ∈ 𝐵)
11		simp23 1209	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵)
12	1, 3	latjcom 18412	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
13	7, 10, 11, 12	syl3anc 1373	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
14	13	breq2d 5121	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
15		simp21 1207	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴)
16	1, 4	atbase 39277	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
17	15, 16	syl 17	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵)
18	1, 3	latjcom 18412	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∨ 𝑋) = (𝑋 ∨ 𝑃))
19	7, 17, 11, 18	syl3anc 1373	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ∨ 𝑋) = (𝑋 ∨ 𝑃))
20	19	breq2d 5121	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑄 ≤ (𝑃 ∨ 𝑋) ↔ 𝑄 ≤ (𝑋 ∨ 𝑃)))
21	5, 14, 20	3imtr4d 294	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑄 ∨ 𝑋) → 𝑄 ≤ (𝑃 ∨ 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  lecple 17233  joincjn 18278  Latclat 18396  Atomscatm 39251  CvLatclc 39253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-lub 18311  df-join 18313  df-lat 18397  df-ats 39255  df-atl 39286  df-cvlat 39310

This theorem is referenced by:  hlexch2  39372