Theorem hlsuprexch 39827

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 2736	. . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾)
4		hlsuprexch.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		eqid 2736	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
6		eqid 2736	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
7		hlsuprexch.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	ishlat2 39799	. . . 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 771	. . . 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 217	. . 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 2994	. . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦))
12		neeq2 2995	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≠ 𝑥 ↔ 𝑧 ≠ 𝑃))
13		oveq1 7374	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ∨ 𝑦) = (𝑃 ∨ 𝑦))
14	13	breq2d 5097	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ≤ (𝑥 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑦)))
15	12, 14	3anbi13d 1441	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((𝑧 ≠ 𝑥 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑥 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦))))
16	15	rexbidv 3161	. . . . . 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 5088	. . . . . . . . 9 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ 𝑧 ↔ 𝑃 ≤ 𝑧))
19	18	notbid 318	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (¬ 𝑥 ≤ 𝑧 ↔ ¬ 𝑃 ≤ 𝑧))
20		breq1 5088	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑥 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑦)))
21	19, 20	anbi12d 633	. . . . . . 7 ⊢ (𝑥 = 𝑃 → ((¬ 𝑥 ≤ 𝑧 ∧ 𝑥 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦))))
22		oveq2 7375	. . . . . . . 8 ⊢ (𝑥 = 𝑃 → (𝑧 ∨ 𝑥) = (𝑧 ∨ 𝑃))
23	22	breq2d 5097	. . . . . . 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 3160	. . . . 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 633	. . . 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 2995	. . . . . 6 ⊢ (𝑦 = 𝑄 → (𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄))
28		neeq2 2995	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≠ 𝑦 ↔ 𝑧 ≠ 𝑄))
29		oveq2 7375	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑃 ∨ 𝑦) = (𝑃 ∨ 𝑄))
30	29	breq2d 5097	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑧 ≤ (𝑃 ∨ 𝑦) ↔ 𝑧 ≤ (𝑃 ∨ 𝑄)))
31	28, 30	3anbi23d 1442	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑦 ∧ 𝑧 ≤ (𝑃 ∨ 𝑦)) ↔ (𝑧 ≠ 𝑃 ∧ 𝑧 ≠ 𝑄 ∧ 𝑧 ≤ (𝑃 ∨ 𝑄))))
32	31	rexbidv 3161	. . . . . 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 7375	. . . . . . . . 9 ⊢ (𝑦 = 𝑄 → (𝑧 ∨ 𝑦) = (𝑧 ∨ 𝑄))
35	34	breq2d 5097	. . . . . . . 8 ⊢ (𝑦 = 𝑄 → (𝑃 ≤ (𝑧 ∨ 𝑦) ↔ 𝑃 ≤ (𝑧 ∨ 𝑄)))
36	35	anbi2d 631	. . . . . . 7 ⊢ (𝑦 = 𝑄 → ((¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑦)) ↔ (¬ 𝑃 ≤ 𝑧 ∧ 𝑃 ≤ (𝑧 ∨ 𝑄))))
37		breq1 5088	. . . . . . 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 3160	. . . . 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 633	. . . 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 3575	. . 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 506	. 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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  ltcplt 18274  joincjn 18277  0.cp0 18387  1.cp1 18388  CLatccla 18464  OMLcoml 39621  Atomscatm 39709  AtLatcal 39710  HLchlt 39796

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-cvlat 39768  df-hlat 39797

This theorem is referenced by:  hlsupr  39832