Theorem cvlexch1 39437

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 39432	. . . . 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 5092	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑥 ↔ 𝑃 ≤ 𝑥))
8	7	notbid 318	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑥))
9		breq1 5092	. . . . . . 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 7354	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑥 ∨ 𝑝) = (𝑥 ∨ 𝑃))
12	11	breq2d 5101	. . . . . 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 7354	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑥 ∨ 𝑞) = (𝑥 ∨ 𝑄))
15	14	breq2d 5101	. . . . . . 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 5092	. . . . . 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 5093	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ 𝑥 ↔ 𝑃 ≤ 𝑋))
20	19	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ 𝑃 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑋))
21		oveq1 7353	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑄) = (𝑋 ∨ 𝑄))
22	21	breq2d 5101	. . . . . . 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 7353	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑃) = (𝑋 ∨ 𝑃))
25	24	breq2d 5101	. . . . . 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 3588	. . . 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 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  joincjn 18217  Atomscatm 39372  AtLatcal 39373  CvLatclc 39374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-cvlat 39431

This theorem is referenced by:  cvlexch2  39438  cvlexchb1  39439  cvlexch3  39441  cvlcvr1  39448  hlexch1  39491