Theorem cvlexch1 39957

Description: An atomic covering lattice has the exchange property. (Contributed by NM, 6-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
cvlexch1	⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Proof of Theorem cvlexch1

Dummy variables 𝑞 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvlexch.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2		cvlexch.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
3		cvlexch.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
4		cvlexch.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	iscvlat 39952	. . . . 5 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))
6	5	simprbi 501	. . . 4 ⊢ (𝐾 ∈ CvLat → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)))
7		breq1 5105	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑥 ↔ 𝑃 ≤ 𝑥))
8	7	notbid 320	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑥))
9		breq1 5105	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑥 ∨ 𝑞) ↔ 𝑃 ≤ (𝑥 ∨ 𝑞)))
10	8, 9	anbi12d 641	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) ↔ (¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞))))
11		oveq2 7406	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑥 ∨ 𝑝) = (𝑥 ∨ 𝑃))
12	11	breq2d 5114	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑞 ≤ (𝑥 ∨ 𝑝) ↔ 𝑞 ≤ (𝑥 ∨ 𝑃)))
13	10, 12	imbi12d 346	. . . . 5 ⊢ (𝑝 = 𝑃 → (((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)) ↔ ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑃))))
14		oveq2 7406	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑥 ∨ 𝑞) = (𝑥 ∨ 𝑄))
15	14	breq2d 5114	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑥 ∨ 𝑞) ↔ 𝑃 ≤ (𝑥 ∨ 𝑄)))
16	15	anbi2d 639	. . . . . 6 ⊢ (𝑞 = 𝑄 → ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) ↔ (¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄))))
17		breq1 5105	. . . . . 6 ⊢ (𝑞 = 𝑄 → (𝑞 ≤ (𝑥 ∨ 𝑃) ↔ 𝑄 ≤ (𝑥 ∨ 𝑃)))
18	16, 17	imbi12d 346	. . . . 5 ⊢ (𝑞 = 𝑄 → (((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) → 𝑄 ≤ (𝑥 ∨ 𝑃))))
19		breq2 5106	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ 𝑥 ↔ 𝑃 ≤ 𝑋))
20	19	notbid 320	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ 𝑃 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑋))
21		oveq1 7405	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑄) = (𝑋 ∨ 𝑄))
22	21	breq2d 5114	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ (𝑥 ∨ 𝑄) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
23	20, 22	anbi12d 641	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) ↔ (¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))))
24		oveq1 7405	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑃) = (𝑋 ∨ 𝑃))
25	24	breq2d 5114	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑄 ≤ (𝑥 ∨ 𝑃) ↔ 𝑄 ≤ (𝑋 ∨ 𝑃)))
26	23, 25	imbi12d 346	. . . . 5 ⊢ (𝑥 = 𝑋 → (((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) → 𝑄 ≤ (𝑥 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))))
27	13, 18, 26	rspc3v 3599	. . . 4 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)) → ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))))
28	6, 27	mpan9 514	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
29	28	exp4b 434	. 2 ⊢ (𝐾 ∈ CvLat → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))))
30	29	3imp 1124	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  lecple 17295  joincjn 18345  Atomscatm 39892  AtLatcal 39893  CvLatclc 39894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-cvlat 39951

This theorem is referenced by:  cvlexch2  39958  cvlexchb1  39959  cvlexch3  39961  cvlcvr1  39968  hlexch1  40011