Theorem hlsuprexch 38247

Description: A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)

Hypotheses

Ref	Expression
hlsuprexch.b	⊢ 𝐵 = (Base‘𝐾)
hlsuprexch.l	⊢ ≤ = (le‘𝐾)
hlsuprexch.j	⊢ ∨ = (join‘𝐾)
hlsuprexch.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
hlsuprexch	⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐾   𝑧,𝑃   𝑧,𝑄

Allowed substitution hints:   ∨ (𝑧)   ≤ (𝑧)

Proof of Theorem hlsuprexch

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlsuprexch.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		hlsuprexch.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		eqid 2732	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		hlsuprexch.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		eqid 2732	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2732	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		hlsuprexch.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat2 38218	. . . 4 ⊢ (𝐾 ∈ HL ↔ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (((0.‘𝐾)(lt‘𝐾)𝑥 ∧ 𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧 ∧ 𝑧(lt‘𝐾)(1.‘𝐾))))))
9		simprl 769	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ∧ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (((0.‘𝐾)(lt‘𝐾)𝑥 ∧ 𝑥(lt‘𝐾)𝑦) ∧ (𝑦(lt‘𝐾)𝑧 ∧ 𝑧(lt‘𝐾)(1.‘𝐾))))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))))
10	8, 9	sylbi 216	. . 3 ⊢ (𝐾 ∈ HL → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))))
11		neeq1 3003	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
12		neeq2 3004	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≠ 𝑥 ↔ 𝑧 ≠ 𝑃))
13		oveq1 7415	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∨ 𝑦) = (𝑃 ∨ 𝑦))
14	13	breq2d 5160	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≤ (𝑥 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑦)))
15	12, 14	3anbi13d 1438	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
16	15	rexbidv 3178	. . . . . 6 ⊢ (𝑥 = 𝑃 → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
17	11, 16	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑃 → ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ↔ (𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)))))
18		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑧 ↔ 𝑃 ≤ 𝑧))
19	18	notbid 317	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑃 ≤ 𝑧))
20		breq1 5151	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑦)))
21	19, 20	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦))))
22		oveq2 7416	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ∨ 𝑥) = (𝑧 ∨ 𝑃))
23	22	breq2d 5160	. . . . . . 7 ⊢ (𝑥 = 𝑃 → (𝑦 ≤ (𝑧 ∨ 𝑥) ↔ 𝑦 ≤ (𝑧 ∨ 𝑃)))
24	21, 23	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑃 → (((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)) ↔ ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))))
25	24	ralbidv 3177	. . . . 5 ⊢ (𝑥 = 𝑃 → (∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥)) ↔ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))))
26	17, 25	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝑃 → (((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) ↔ ((𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)))))
27		neeq2 3004	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
28		neeq2 3004	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≠ 𝑦 ↔ 𝑧 ≠ 𝑄))
29		oveq2 7416	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
30	29	breq2d 5160	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≤ (𝑃 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑄)))
31	28, 30	3anbi23d 1439	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
32	31	rexbidv 3178	. . . . . 6 ⊢ (𝑦 = 𝑄 → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
33	27, 32	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑄 → ((𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))) ↔ (𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄)))))
34		oveq2 7416	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
35	34	breq2d 5160	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑃 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑄)))
36	35	anbi2d 629	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄))))
37		breq1 5151	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦 ≤ (𝑧 ∨ 𝑃) ↔ 𝑄 ≤ (𝑧 ∨ 𝑃)))
38	36, 37	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑄 → (((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)) ↔ ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
39	38	ralbidv 3177	. . . . 5 ⊢ (𝑦 = 𝑄 → (∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃)) ↔ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
40	33, 39	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝑄 → (((𝑃 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑃))) ↔ ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃)))))
41	26, 40	rspc2v 3622	. . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ≠ 𝑦 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) → 𝑦 ≤ (𝑧 ∨ 𝑥))) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃)))))
42	10, 41	mpan9 507	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))
43	42	3impb 1115	1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ≠ 𝑄 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))) ∧ ∀𝑧 ∈ 𝐵 ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄)) → 𝑄 ≤ (𝑧 ∨ 𝑃))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   class class class wbr 5148  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  ltcplt 18260  joincjn 18263  0.cp0 18375  1.cp1 18376  CLatccla 18450  OMLcoml 38040  Atomscatm 38128  AtLatcal 38129  HLchlt 38215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-cvlat 38187  df-hlat 38216

This theorem is referenced by:  hlsupr  38252