Theorem cvlexch1 39321

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 39316	. . . . 5 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))
6	5	simprbi 496	. . . 4 ⊢ (𝐾 ∈ CvLat → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)))
7		breq1 5110	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑥 ↔ 𝑃 ≤ 𝑥))
8	7	notbid 318	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑥))
9		breq1 5110	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑥 ∨ 𝑞) ↔ 𝑃 ≤ (𝑥 ∨ 𝑞)))
10	8, 9	anbi12d 632	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) ↔ (¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞))))
11		oveq2 7395	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑥 ∨ 𝑝) = (𝑥 ∨ 𝑃))
12	11	breq2d 5119	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑞 ≤ (𝑥 ∨ 𝑝) ↔ 𝑞 ≤ (𝑥 ∨ 𝑃)))
13	10, 12	imbi12d 344	. . . . 5 ⊢ (𝑝 = 𝑃 → (((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)) ↔ ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑃))))
14		oveq2 7395	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑥 ∨ 𝑞) = (𝑥 ∨ 𝑄))
15	14	breq2d 5119	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑥 ∨ 𝑞) ↔ 𝑃 ≤ (𝑥 ∨ 𝑄)))
16	15	anbi2d 630	. . . . . 6 ⊢ (𝑞 = 𝑄 → ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) ↔ (¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄))))
17		breq1 5110	. . . . . 6 ⊢ (𝑞 = 𝑄 → (𝑞 ≤ (𝑥 ∨ 𝑃) ↔ 𝑄 ≤ (𝑥 ∨ 𝑃)))
18	16, 17	imbi12d 344	. . . . 5 ⊢ (𝑞 = 𝑄 → (((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) → 𝑄 ≤ (𝑥 ∨ 𝑃))))
19		breq2 5111	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ 𝑥 ↔ 𝑃 ≤ 𝑋))
20	19	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ 𝑃 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑋))
21		oveq1 7394	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑄) = (𝑋 ∨ 𝑄))
22	21	breq2d 5119	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ (𝑥 ∨ 𝑄) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
23	20, 22	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) ↔ (¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))))
24		oveq1 7394	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑃) = (𝑋 ∨ 𝑃))
25	24	breq2d 5119	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑄 ≤ (𝑥 ∨ 𝑃) ↔ 𝑄 ≤ (𝑋 ∨ 𝑃)))
26	23, 25	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑋 → (((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) → 𝑄 ≤ (𝑥 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))))
27	13, 18, 26	rspc3v 3604	. . . 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 506	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
29	28	exp4b 430	. 2 ⊢ (𝐾 ∈ CvLat → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))))
30	29	3imp 1110	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  Atomscatm 39256  AtLatcal 39257  CvLatclc 39258

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-ext 2701

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-cvlat 39315

This theorem is referenced by:  cvlexch2  39322  cvlexchb1  39323  cvlexch3  39325  cvlcvr1  39332  hlexch1  39376