Theorem cvlexch1 38198

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 38193	. . . . 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 5152	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑥 ↔ 𝑃 ≤ 𝑥))
8	7	notbid 318	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑥))
9		breq1 5152	. . . . . . 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 7417	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑥 ∨ 𝑝) = (𝑥 ∨ 𝑃))
12	11	breq2d 5161	. . . . . 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 7417	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑥 ∨ 𝑞) = (𝑥 ∨ 𝑄))
15	14	breq2d 5161	. . . . . . 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 5152	. . . . . 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 5153	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑃 ≤ 𝑥 ↔ 𝑃 ≤ 𝑋))
20	19	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ 𝑃 ≤ 𝑥 ↔ ¬ 𝑃 ≤ 𝑋))
21		oveq1 7416	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑄) = (𝑋 ∨ 𝑄))
22	21	breq2d 5161	. . . . . . 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 7416	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ 𝑃) = (𝑋 ∨ 𝑃))
25	24	breq2d 5161	. . . . . 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 3628	. . . 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 508	. . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) → ((¬ 𝑃 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → 𝑄 ≤ (𝑋 ∨ 𝑃)))
29	28	exp4b 432	. 2 ⊢ (𝐾 ∈ CvLat → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))))
30	29	3imp 1112	1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑃 ≤ 𝑋) → (𝑃 ≤ (𝑋 ∨ 𝑄) → 𝑄 ≤ (𝑋 ∨ 𝑃)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  Atomscatm 38133  AtLatcal 38134  CvLatclc 38135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-cvlat 38192

This theorem is referenced by:  cvlexch2  38199  cvlexchb1  38200  cvlexch3  38202  cvlcvr1  38209  hlexch1  38253