Theorem cvlexch1 39829

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 39824	. . . . 5 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))
6	5	simprbi 498	. . . 4 ⊢ (𝐾 ∈ CvLat → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)))
7		breq1 5076	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑥 ↔ 𝑃 ≤ 𝑥))
8	7	notbid 319	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑥))
9		breq1 5076	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ (𝑥 ∨ 𝑞) ↔ 𝑃 ≤ (𝑥 ∨ 𝑞)))
10	8, 9	anbi12d 638	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) ↔ (¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞))))
11		oveq2 7365	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑥 ∨ 𝑝) = (𝑥 ∨ 𝑃))
12	11	breq2d 5085	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑞 ≤ (𝑥 ∨ 𝑝) ↔ 𝑞 ≤ (𝑥 ∨ 𝑃)))
13	10, 12	imbi12d 345	. . . . 5 ⊢ (𝑝 = 𝑃 → (((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝)) ↔ ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑃))))
14		oveq2 7365	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑥 ∨ 𝑞) = (𝑥 ∨ 𝑄))
15	14	breq2d 5085	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑥 ∨ 𝑞) ↔ 𝑃 ≤ (𝑥 ∨ 𝑄)))
16	15	anbi2d 636	. . . . . 6 ⊢ (𝑞 = 𝑄 → ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) ↔ (¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄))))
17		breq1 5076	. . . . . 6 ⊢ (𝑞 = 𝑄 → (𝑞 ≤ (𝑥 ∨ 𝑃) ↔ 𝑄 ≤ (𝑥 ∨ 𝑃)))
18	16, 17	imbi12d 345	. . . . 5 ⊢ (𝑞 = 𝑄 → (((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) → 𝑄 ≤ (𝑥 ∨ 𝑃))))
19		breq2 5077	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ 𝑥 ↔ 𝑃 ≤ 𝑋))
20	19	notbid 319	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ 𝑃 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑋))
21		oveq1 7364	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑄) = (𝑋 ∨ 𝑄))
22	21	breq2d 5085	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ (𝑥 ∨ 𝑄) ↔ 𝑃 ≤ (𝑋 ∨ 𝑄)))
23	20, 22	anbi12d 638	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) ↔ (¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))))
24		oveq1 7364	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑃) = (𝑋 ∨ 𝑃))
25	24	breq2d 5085	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑄 ≤ (𝑥 ∨ 𝑃) ↔ 𝑄 ≤ (𝑋 ∨ 𝑃)))
26	23, 25	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑋 → (((¬ 𝑃 ≤ 𝑥 ∧ 𝑃 ≤ (𝑥 ∨ 𝑄)) → 𝑄 ≤ (𝑥 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃))))
27	13, 18, 26	rspc3v 3576	. . . 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 511	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
29	28	exp4b 431	. 2 ⊢ (𝐾 ∈ CvLat → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))))
30	29	3imp 1116	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   class class class wbr 5073  ‘cfv 6486  (class class class)co 7357  Basecbs 17171  lecple 17219  joincjn 18269  Atomscatm 39764  AtLatcal 39765  CvLatclc 39766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-iota 6442  df-fv 6494  df-ov 7360  df-cvlat 39823

This theorem is referenced by:  cvlexch2  39830  cvlexchb1  39831  cvlexch3  39833  cvlcvr1  39840  hlexch1  39883