Theorem cvlexch3 40084

Description: An atomic covering lattice has the exchange property. (atexch 32711 analog.) (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
cvlexch3.b	⊢ 𝐵 = (Base‘𝐾)
cvlexch3.l	⊢ ≤ = (le‘𝐾)
cvlexch3.j	⊢ ∨ = (join‘𝐾)
cvlexch3.m	⊢ ∧ = (meet‘𝐾)
cvlexch3.z	⊢ 0 = (0.‘𝐾)
cvlexch3.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cvlexch3	⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Proof of Theorem cvlexch3

Step	Hyp	Ref	Expression
1		cvlatl 40077	. . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat)
2	1	adantr 485	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝐾 ∈ AtLat)
3		simpr1 1213	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑃 ∈ 𝐴)
4		simpr3 1215	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
5		cvlexch3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
6		cvlexch3.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
7		cvlexch3.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
8		cvlexch3.z	. . . . 5 ⊢ 0 = (0.‘𝐾)
9		cvlexch3.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
10	5, 6, 7, 8, 9	atnle 40069	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
11	2, 3, 4, 10	syl3anc 1398	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
12		cvlexch3.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
13	5, 6, 12, 9	cvlexch1 40080	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
14	13	3expia 1139	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))))
15	11, 14	sylbird 263	. 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ∧ 𝑋) = 0 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃))))
16	15	3impia 1135	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  joincjn 18368  meetcmee 18369  0.cp0 18478  Atomscatm 40015  AtLatcal 40016  CvLatclc 40017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18351  df-poset 18370  df-plt 18385  df-lub 18401  df-glb 18402  df-join 18403  df-meet 18404  df-p0 18480  df-lat 18489  df-covers 40018  df-ats 40019  df-atl 40050  df-cvlat 40074

This theorem is referenced by:  hlexch3  40143